Method and system for determining wind chill temperature

ABSTRACT

The present invention includes methods, systems and computer-readable media for more accurately determining wind chill temperature, T wc , equivalent temperature, T eq , time to freeze, t f , facial temperature, T f   m+Δt , as a function of time and the altitude correction factor, Δt f /1000. The wind chill model of the present invention accounts for the two major heat losses (forced convection, radiation) and a minor heat loss (evaporative cooling) from the facial surface and is also capable of accounting for the two major heat gains (metabolic, solar) at the facial surface due to the individual&#39;s physical activity and the presence of sunshine. The wind chill model of the present invention also provides a more accurate value for the wind velocity at head level.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is a divisional patent application of U.S. nonprovisional patentapplication Ser. No. 11/481,684, filed Jul. 6, 2006, now U.S. Pat. No.7,481,576 titled: “METHOD AND SYSTEM FOR DETERMINING WIND CHILLTEMPERATURE”, the contents of which are herein incorporated by referencefor all purposes.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to methods and systems for determiningwind chill temperature. More specifically, the invention includesmethods and systems for determining the wind chill temperature and thetime to freeze for the facial temperature.

2. Description of Related Art

The sensation upon exposed skin that the ambient temperature in thepresence of wind feels colder than the actual measured temperature isoften referred to as the wind chill temperature. The concept of “windchill” was first proposed by Siple, P. A., and C. F. Passel,“Measurements of dry atmospheric cooling in subfreezing temperatures”,Proc. Amer. Phil. Soc., Vol. 89, No. 1, pp. 177-199, 1945. The Siple andPassel experiments and model have been criticized by investigators asbeing primitive, flawed and lacking a theoretical basis. However, theSiple and Passel experiments yielded results that have proven useful forsix decades. One criticism of the Siple and Passel model is that itincorrectly assumes a constant skin temperature of 33° C. (91.4° F.)during the entire time of skin exposure. This assumption was known toresult in predicted values of the wind chill temperatures that would becolder than the actual values.

Despite this criticism, Osczevski, R. J., “The basis of wind chill”,Arctic., Vol. 48, No. 4, pp. 372-382, 1995, gave credence to thesepredictions when he stated that the test cylinder used by Siple andPassel was nearly the perfect size to represent the human head. This mayexplain, at least partially, why these predictions have served so wellover the intervening years.

Wind chill temperature is an actual temperature that it is notrestricted to the winter season. The wind chill temperature sensed by anindividual on a windy, cold winter day is, conceptually, no differentthan that which he senses in front of an electric fan on a hot summerday. From a health or safety standpoint, the wind chill temperaturesensed in the winter is the one which causes concern. The reason forthis is the subjective nature of this temperature, i.e., the temperaturesensed by one person may be quite different from that sensed by another.This is believed to be especially true at very low temperatures where anoticeable difference between the actual and perceived temperatures maybe quite difficult.

Individual differences in sensing these low temperatures have led toproposed solutions on how the subjective nature of this temperaturecould be minimized. Suggestions have been made that the wind chilltemperature should be replaced with categories such as “cold”, “verycold” and “extremely cold” or combining it with the heat index to comeup with comfort index categories ranging from “minus to plus 10”.However, these approaches are also very subjective. Fortunately, neitherof these proposed solutions has prevailed.

The subjective nature of the wind chill temperature has led tosuggestions that it should be combined with clothing distributions todefine a comfort level to which an individual can more easily relate.Because the comfort level is determined by the warmer temperaturessensed by the clothed surface of the body, this approach could mask apotentially dangerous situation. This could occur if the individualfeels so comfortable, as a result of being adequately dressed, that heis unaware that his face may be subjected to the most imminent hazard ofextended exposure, i.e., frostbite.

A more recent development is the wind chill model disclosed inBluestein, M. and Zecher, J., “A new approach to an accurate wind chillfactor”, Bull. Amer. Meteor. Soc., Vol. 80, No. 9, pp. 1893-1899, 1999.The National Oceanic & Atmospheric Administration's National WeatherService adopted the Bluestein and Zecher wind chill model on Nov. 1,2001. Developed as an analytical counterpart to the Siple and Passelexperiment, the Bluestein and Zecher model corrected for the constantskin temperature assumption by allowing it to vary, i.e., decrease withincreasing exposure time. This was expected to give correct wind chilltemperatures that were warmer than the Siple and Passel values. Inaddition, Bluestein and Zecher used a wind speed reduction at head levelbased upon the assumption that the free-stream velocity is always 50%greater. With this assumption and the skin temperature correction, theBluestein and Zecher model does indeed predict wind chill temperaturesthat are as much as 15° F. (8.33° C.) warmer than the correspondingSiple and Passel values. However, a close examination of the Bluesteinand Zecher results shows that essentially all of this warming is due tothe wind reduction at head level with, at most, 2° F. (1.1° C.) beingdue to the varying skin temperature. At very low temperatures and highvelocities the Bluestein and Zecher results show no moderationwhatsoever. Instead of a moderation, their wind chill temperatures areapproximately 1° F. (0.56° C.) colder than the Siple and Passel values.This result calls into question the accuracy of the Bluestein and Zechermodel. Unfortunately, the Bluestein and Zecher model assumes that thefree-stream velocity is always 50% greater than that at head level.Boundary layer calculations for all individuals in real life situationsshow that this 33% reduction in the velocity at head level is a uniquecondition which will almost never exist; in fact analyses will show thatin all instances the reduction will be at or near zero. Without thisincorrect velocity reduction, Bluestein and Zecher's results areactually no different than those of Siple and Passel.

Various devices have been proposed for determining wind chilltemperature using the conventional methods disclosed by Siple andPassel, Bluestein and Zecher and others. Such conventional wind chilltemperature devices are disclosed in U.S. Pat. No. 3,753,371 toAnderson, U.S. Pat. No. 3,954,007 to Harrigan, U.S. Pat. No. 4,047,431to Mulvaney et al., U.S. Pat. No. 4,106,339 to Baer, U.S. Pat. No.4,261,201 to Howard and PCT Patent Application No. WO 81/00462 toHoward. Generally, these devices are based on measurements of airtemperature and wind speed only. Moreover, none of these conventionaldevices appears to correct for the above-noted errors in the prior artmethods. Furthermore, the inventors are not aware of any methods orsystems that account for other important factors such as altitude,insolation, and metabolic heat generation.

For the above reasons, it would be highly advantageous to provide a moreaccurate and complete wind chill model. It would also be advantageous toprovide a system and method for wind chill determination based on a moreaccurate and complete wind chill model.

SUMMARY OF THE INVENTION

The present invention includes methods for more accurately determiningwind chill temperature, T_(wc), equivalent temperature, T_(eq), time tofreeze, t_(f), facial temperature, T_(f) ^(m+Δt), as a function of timeand the altitude correction factor, Δt_(f)/1000. Methods for determininga wind reduction factor, WRF, are also disclosed.

Embodiments of computer-readable media storing computer-executableinstructions for performing the various methods of the present inventionare also disclosed.

A system for implementing one or more of the methods of determining windchill temperature, T_(wc), equivalent temperature, T_(eq), time tofreeze, t_(f), facial temperature, T_(f) ^(m+Δt), as a function of timeand the altitude correction factor, Δt_(f)/1000, according to thepresent invention is also disclosed.

Additional features and advantages of the invention will be set forth inthe description which follows, and in part will be apparent from thedescription, or may be learned by the practice of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The following drawings illustrate exemplary embodiments for carrying outthe invention. Like reference numerals refer to like parts in differentviews or embodiments of the present invention in the drawings.

FIG. 1A is a diagram of a two-dimensional segment used in thedetermination of the equivalent temperature (T_(eq)), heat losses andgains on the segment, according to the present invention.

FIG. 1B is a diagram of the two-dimensional segment shown in FIG. 1Aillustrating equivalent heat loss (q_(nc)) on the segment when V=0,according to the present invention.

FIG. 2 is a diagram of a segment of cylindrical facial surface used inthe determination of the wind chill temperature (T_(wc)), according tothe present invention.

FIGS. 3A and 3B are graphs of the Biot number, Bi, as a function of windvelocity, V, and the Fourier number, F₀, plotted as a function ofexposure time, t, respectively.

FIGS. 4A and 4B are graphs of facial temperature (T_(f)) decay curvesand time to freeze (t_(f)) as a function of wind velocity.

FIG. 5 is a graph of time to freeze (t_(f)) as a function of ambientconditions (T_(a), V) and q_(e)=q_(b)=q_(i)=0.

FIG. 6A is a graph comparing the forced convection coefficients as theyapply to the Siple and Passel container.

FIG. 6B is a graph comparing the Siple and Passel wind chilltemperatures with those predicted by the wind chill model of the presentinvention using the Siple and Passel container.

FIG. 7A is a graph comparing the predicted forced convectioncoefficients for the head with those of Siple and Passel.

FIG. 7B is a graph comparing the Siple and Passel wind chilltemperatures with those predicted by the wind chill model of the presentinvention for the simulated head.

FIG. 8 is a graph comparing the Bluestein and Zecher results with thoseof Siple and Passel.

FIG. 9 is a graph of experimental data on facial temperatures (T_(f)) asa function of ambient temperature and wind velocity.

FIG. 10 is a graph comparing facial temperature (T_(f)) variation usingEq. (22) according to the present invention with the prior art Buettnerresults.

FIG. 11 is a graph comparing temperatures on the facial components asfound by LeBlanc et al. of the prior art with the facial temperatures(T_(f)) as calculated from Eq. (22) according to the present invention.

FIGS. 12A-C are graphs of the surface temperature (T_(s)) of a simulatedhead in home freezer experiments using a water-filled wood bowl,chicken, and pot roast, respectively.

FIG. 13 is a graph of facial temperatures (T_(f)) as predicted from Eq.(22) with the Discovery Channel™ dexterity test results.

FIGS. 14A-B are graphs comparing the facial temperatures (T_(f)) aspredicted from Eq. (22) with the prior art Adamenko and Khairullin testresults for G=5 Btu/hr-ft² (15.77 W/m²) (FIG. 14A) and G=7.8 Btu/hr-ft²(24.61 W/m²) (FIG. 14B).

FIG. 15 is a graph of facial temperature (T_(f)) and wind chilltemperature (T_(wc)) as a function of exposure time.

FIG. 16 is a graph of facial temperature (T_(f)) decay for skiers in theWorld Downhill Ski Championship 2004.

FIG. 17 is a graph for determining the actual time to freeze (t_(f)) fora Minneapolis weather forecast.

FIG. 18 is a graph of facial temperature (T_(f)) decay and time tofreeze (t_(f)) as a function of velocity in Antarctica.

FIG. 19A-B are graphs of the effect of heat losses on the predicted windchill temperature (T_(wc)) sustained by a Boston resident casuallywalking on a sunny, cold winter day under the following conditions: H=0ft, G=42.66 Btu/hr-ft² (134.58 W/m²), M_(act)=31 Btu/hr-ft² (97.79W/m²), w=0.00655 lbm/hr-ft² (0.0320 kg/hr-r²), WRF=0.

FIG. 20A-B are graphs of the effect of heat gains on the predicted windchill temperature (T_(wc)) sustained by a Boston resident casuallywalking on a sunny, cold winter day under the following conditions: H=0ft, G=42.66 Btu/hr-ft² (134.58 W/m²), M_(act)=31 Btu/hr-ft² (97.79W/m²), w=0.00655 lbm/hr-ft² (0.0320 kg/hr-m²), WRF=0.

FIGS. 21A-B are graphs of the effects of heat losses and heat gains onthe facial temperature (T_(f)) decay and the time to freeze (t_(f)).

FIGS. 22A-B are graphs of the effects of heat losses and heat gains onthe facial temperature (T_(f)) decay and the time to freeze (t_(f)).

FIGS. 23A-B are graphs of the effects of heat losses and heat gains onthe facial temperature (T_(f)) decay and the time to freeze (t_(f))

FIG. 24 is a graph of facial temperature (T_(f)) decay and the time tofreeze (t_(f)) using the infinite series solution of Eq. (23h).

FIGS. 25A-B are graphs of wind chill temperature as a function ofaltitude when T_(a)=0° F. (−17.78° C.) and T_(a)=−40° F. (−40° C.),respectively.

FIG. 26 is a graph of wind chill temperature/altitude gradient as afunction of ambient conditions.

FIG. 27 is a graph of facial temperature decay and time to freeze(t_(f)) as a function of altitude.

FIG. 28 is a graph of incremental time to freeze/altitude gradient as afunction of ambient conditions.

FIG. 29 is a graph of wind chill temperature (T_(wc)), facialtemperature (T_(f)) and time to freeze (t_(f)) during extreme weatherconditions on Mt. Washington.

FIG. 30 is a diagram illustrating a wind-generated turbulent boundarylayer according to the present invention.

FIG. 31 is a diagram illustrating turbulent boundary layer thickness (δ)as a function of wind speed and wind/surface contact distance (l)according to the present invention.

FIG. 32 is a graph illustrating maximum wind/surface contact distance(l_(max)) and obstruction distance (D) for a turbulent boundary layerthickness (δ) of 5 ft (1.52 m).

FIG. 33 is a graph illustrating wind reduction factor as a function ofwind speed and distance (D) using Steadman's exponent.

FIG. 34 is a graph of wind reduction factor as a function of theexponent denominator when (y/δ)=0.5.

FIG. 35 is a block diagram of an embodiment of a computer-readablemedium suitable for storing computer-executable instructions forcalculating wind chill temperature, T_(wc), equivalent temperature,T_(eq), and time to freeze, t_(f), and any other related calculationsaccording to the present invention.

FIG. 36 is a system for determining wind chill temperature, T_(wc),equivalent temperature, T_(eq), and time to freeze, t_(f), and any otherrelated calculations according to the present invention.

DETAILED DESCRIPTION

The wind chill model disclosed herein is confined to the facial surfacesince the face of even a properly dressed individual will most likely beuncovered. Furthermore, the individual senses the wind chill temperatureby way of the sensory nerves located within the intermediate skin layer(dermis) of the exposed facial surface and not from the surface of theclothed body. The wind chill model disclosed herein consists of anequation for the wind chill temperature and a second equation called thetime to freeze. The latter defines the length of the exposure time atthe wind chill temperature when facial freezing occurs. The time tofreeze feature eliminates much of the controversy associated with windchill by reducing the subjective nature of the wind chill temperatureand at the same time provide a warning to the individual of a safeexposure time.

The wind chill model disclosed herein may be viewed as a complete modelin that it is not restricted to the two major heat losses (forcedconvection, radiation) from the facial surface but is also capable ofaccounting for the two major heat gains (metabolic, solar) at thesurface due to the individual's physical activity and the presence ofsunshine. A lesser heat loss (evaporation) resulting from the physicalactivity is also included. The model incorporates two changes in theforced convection heat loss expression. One aspect of wind chill thathas escaped attention in the prior art is the dependency of this heatloss on the ambient pressure and consequently altitude. The reason forthis may have been due to both oversight and the belief that the effectof pressure was negligible. However, such is not the case and thepresent model shows that this effect can be quite significant. Sincealtitude, rather than ambient pressure, will generally be the knownquantity in wind chill calculations, the forced convection term wasexpressed in terms of altitude. The second change involves defining anaccurate value for the wind velocity at head level. Bluestein and Zechercorrectly noted that this velocity might be less than that at the NWS 10m level, but incorrectly used a value that is referred to in this modelas the wind reduction factor (WRF). Their value of WRF was 0.33 basedupon the velocity profile acquired in a test by Steadman (1971). ThisWRF reflects their assumption that the NWS 10 m velocity would always be50% greater than the velocity at head level. Bluestein and Zecher madethe error of assuming the boundary layer profile at the Steadman testsite as being the same for all individuals at all times irrespective oftheir location. Clearly, this is not so since the wind-generatedboundary layer adjacent to an individual is dependent upon individual'sdistance from the nearest wind obstruction. The wind chill model of thepresent invention, which provides a procedure for determining the WRFfor each specific case, uses the WRF as a multiple of the NWS 10 mvelocity value in the forced convection heat loss term.

This model, which has been validated experimentally, is believed toprovide the most accurate prediction of the wind chill temperature andthe only means of predicting the corresponding time to freeze. It isapplicable to all individuals engaged in a wide range of physicalactivity in either the presence or absence of sunshine. Finally, themodel is adaptable over a wide range of ambient temperatures, windvelocities, altitudes, and geographical locations worldwide.

Development of the Model Equations: The model consists of twoanalytically derived equations, one for the wind chill temperature, andone for the time to freeze. Each of these equations was derived from aprevious equation. Because wind chill is not restricted to humans butapplies to inanimate objects as well, the wind chill temperatureequation was obtained from its basic counterpart, the equivalenttemperature for objects. The time to freeze equation is a curve fit ofextensive results from a time dependant facial temperature equation inwhich facial freezing times were determined over a wide range of ambientconditions. The equations are described below in the order in which theywere developed.

Equivalent Temperature Equation: To really understand the meaning andsignificance of the equivalent temperature and appreciate the scope ofits application, its expression was derived analytically. This was donethrough the use of the two-dimensional surface segment as shown in FIGS.1A-B. In FIG. 1A, the right side of the segment is exposed to a windvelocity V and an ambient temperature T_(a) which is presumed to be lessthan the right surface temperature T_(s). This wind results in a forcedconvection heat loss (q_(fc)) from the segment. If this same heat losswas assumed to exist in the absence of the wind, the right surface wouldbe “sensing” a temperature less than T_(a). This sensed temperature isby definition the equivalent temperature (T_(eq)) in its most basicform. But since T_(s)>T_(a), there is also a radiation heat loss (q_(r))and possibly evaporation heat loss (q_(e)) from the surface. These twoheat losses augment the forced convection heat loss so as to decreasethe T_(eq). However, the right side being exposed to the ambienttemperature T_(a) may also receive heat due to solar radiation orinsolation (q_(i)). In the case where the left surface temperatureT_(b)>T_(a), the right surface will also receive heat (q_(b)) due toconduction through the segment. Both of these heat gains (q_(i), q_(b))have a moderating effect on T_(eq) and will tend to increase its value.

FIG. 1B shows the summation of all these heat losses and gains into onenatural convection heat loss (q_(nc)) when V=0. From the temperaturedifference (T_(s)−T_(eq)) that results in this natural convection heatloss, one can determine T_(eq). This was the procedure used in thedevelopment of this model. It is described in the following paragraphs:

The forced convection heat loss, q_(fc), may be expressed as:q _(fc) =h _(fc)(T _(s) −T _(a))  (1)Where h_(fc) is forced convection heat transfer coefficient and theradiation heat loss, q_(r), is:q _(r=εσ() T _(s) ⁴ −T _(a) ⁴)  (2)where ε is the emissivity of the surface and σ is the Stefan-Boltzmannconstant. The evaporation heat loss, q_(e), for a wetted surface is:q_(e)={dot over (w)}l_(e)  (3)where {dot over (w)} is the water evaporation flux rate and l_(e) is thelatent heat of evaporation of the wetting liquid. The heat addition dueto insolation, q_(i), is:q_(i)=αG  (4)where α is the thermal absorptivity of the surface and G is the localinsolation value. The heat addition due to conduction into the segment,q_(b), is:

$\begin{matrix}{q_{b} = {\frac{k}{s}\left( {T_{b} - T_{s}} \right)}} & (5)\end{matrix}$where k and s are the thermal conductivity and total thickness,respectively, of the segment. The natural convection heat loss, q_(nc),in the absence of velocity is equal to the sum of all the heat lossesand gains as shown in FIG. 1A and is:q _(nc) =q _(fc) +q _(r) +q _(e) −q _(i) −q _(b)  (6)which can now be expressed as follows in terms of the previously definedequivalent temperature (T_(eq)):q _(nc) =h _(nc)(T _(s) −T _(eq))  (7)The natural convection heat transfer coefficient (h_(nc)) disclosed inJakob, M. and Hawkins, G. A., Elements of Heat Transfer and Insulation,John Wily & Sons, p. 107, 1954, can be expressed as:

$\begin{matrix}{h_{nc} = {C_{1}\left( \frac{T_{s} - T_{eq}}{L} \right)}^{\phi}} & \left( {8\; a} \right)\end{matrix}$where the exponent φ=0.25 for a heated vertical plane or cylindricalsurface and where the sole source of heat is within the surface itself.In instances where additional heat is being conducted through thesurface, from inside to outside, φ is expected to take on other values.The coefficient C₁ is a function of air density, thermal conductivity,specific heat, dynamic viscosity and the coefficient of thermalexpansion and where L is the length of the segment. The air density (ρ),dynamic viscosity (μ), specific heat (C_(p)) and thermal conductivity(k) are calculated at the film temperature (T_(film)) which is definedas the average of the surface and air temperatures, that isT_(film)=(T_(s)+T_(a))/2. The expression for C₁ as disclosed in Chapman,A. J. Heat Transfer, 3rd ed., Macmillan, p. 384, 1974, is:

$\begin{matrix}{C_{1} = {0.59\;{k\left( \frac{g\;\beta\;\rho^{2}C_{p}}{\mu\; k} \right)}^{0.25}}} & \left( {8\; b} \right)\end{matrix}$where g is the gravitational constant and the coefficient of thermalexpansion (β) of the air is β=(T_(film))⁻¹. Values of C₁ were calculatedover a wide range of ambient temperatures (−291.4° F.≦T_(a)≦108.6° F.(−179.67° C.≦T_(a)≦42.56° C.)) plotted as a function of the filmtemperature expressed in terms of the surface and ambient temperature(T_(s), T_(a)) and then curve fitted using TableCurve 2D™ to obtain thefollowing expression:C ₁=0.3268−9.5500×10⁻⁵(T _(s) +T _(a))  (8c)and where the correlation coefficient r²=0.9953. TableCurve 2D™ isavailable from Systat Software, Inc., 501 Canal Blvd, Suite E, PointRichmond, Calif.

Substituting Eqs. (1), (2), (3), (4), (5), (7), and (8a) into Eq. (6),and solving for the equivalent temperature, T_(eq), gives:

$\begin{matrix}{T_{eq} = {T_{s} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{{h_{fc}\left( {T_{s} - T_{a}} \right)} + {ɛ\;{\sigma\left( {T_{s}^{4} - T_{a}^{4}} \right)}} +} \\{{\overset{.}{w}l_{e}} - {\alpha\; G} - {\frac{k}{s}\left( {T_{b} - T_{s}} \right)}}\end{bmatrix}} \right\}^{1/{({1 + \phi})}}}} & (9)\end{matrix}$where C₁ is calculated from Eq. (8c). Equation (9) is the most basicequation for the equivalent temperature of a planar surface. Theconductive, insolative and radiative heat transfer quantities (k, s, αand ε) in the equation depend upon the material properties of thesurface while the forced convective heat transfer coefficient (h_(fc))does not. Therefore, by substituting an expression for h_(fc) into thisequation, it can be made applicable to all two-dimensional surfaces.This was done by selecting convective heat transfer coefficients for aflat surface segment disclosed in Harms et al., Aerodynamic Heating ofHigh-Speed Aircraft, Bell Aerosystems Company, Report No. 7006-3352-001,Vol. 1, pp. 14-16, 1959. Harms et al. expresses the incompressible formof the forced convection coefficient (h_(fc)) in terms of the Nusselt(Nu), Reynolds (Re), and molecular Prandtl (Pr) numbers. The forms are:

$\begin{matrix}{{{Nu} = {{h_{fc}\frac{x}{k}} = {0.332\left( {Re}_{x} \right)^{0.5}\left( \Pr \right)^{0.33}}}}{{{for}\mspace{14mu}{laminar}\mspace{14mu}{flow}},{and}}} & \left( {10\; a} \right) \\{{Nu} = {{h_{fc}\frac{x}{k}} = {0.0296\left( {Re}_{x} \right)^{0.8}\left( \Pr \right)^{0.33}}}} & \left( {10\; b} \right)\end{matrix}$for turbulent flow, where the characteristic length (x) in the Re numberis the length L of the segment. Because the surface temperature of thesegment may differ from the free-stream temperature and because of theabsence of a pressure gradient within the surface boundary layer, thedensity varies across the layer and the flow is considered compressible.A solution for the compressible laminar boundary layer is disclosed inCrocco, L., The laminar boundary layer in gases, Translation, NorthAmerican Aviation, Aerophysics Laboratory, AL-684, 1948, while vanDriest, E. R., “Turbulent boundary layer in compressible fluids,” J. ofAero. Sciences, Vol. 18, pp. 145-160, 1951, discloses a similar solutionfor a turbulent boundary layer. Harms et al. describes and uses asuccessful correlation of the Crocco and van Driest results byevaluating all of the transport and fluid properties in Eqs. (10a) and(10b) in terms of the following reference temperature, T^(l), assuggested in Eckert, E. R. G. Survey on heat transmission at highspeeds, USAF Wright Patterson, WADC TR 54-70, 1954:

$\begin{matrix}{T^{l} = {{0.5\left( {T_{s} + T_{a}} \right)} + {0.22\left( \frac{\gamma - 1}{2} \right)M^{2}T_{a}}}} & \left( {10\; c} \right)\end{matrix}$In Eq (10c), the first term on the right side represents a statictemperature component and the second term is a recovery temperature or adynamic temperature component expressed as a function of the Mach number(M). By using T^(l) as the reference temperature, the resulting forcedconvection heat transfer coefficients for laminar flow (h_(fc,l)) andturbulent flow (h_(fc,t)) are applicable to all velocities (V) and allsurface temperatures (T_(s)). For wind chill calculations, wheregenerally M<0.1, the dynamic term in T^(l) can be neglected. Thus, it ispossible to reduce Eqs. (10a) and (10b) to the following forms of theforced convection coefficient:

$\begin{matrix}{h_{{f\; c},l,s} = \frac{0.00963\left( {P\; V} \right)^{0.5}}{\left\lbrack {0.5\left( {T_{s} + T_{a}} \right)} \right\rbrack^{0.04}L^{0.5}}} & \left( {11a} \right)\end{matrix}$for the laminar flow along the segment, and

$\begin{matrix}{h_{{f\; c},t,s} = \frac{0.0334\left( {P\; V} \right)^{0.8}}{\left\lbrack {0.5\left( {T_{s} + T_{a}} \right)} \right\rbrack^{0.576}L^{0.2}}} & \left( {11b} \right)\end{matrix}$for turbulent flow. In these equations P is the ambient pressure, andthe ambient (T_(a)) and surface temperatures (T_(s)) are both expressedin ° F. absolute (or ° R). In this form, values of h_(fc) can be readilyexpressed in terms of the variables P, V, T_(a), and T_(s), which areprimary variables in this study.

Flow along a portion or possibly the entire length of the segment willbe laminar. Substitution of the laminar forced convection coefficient inEq. (11a) into Eq. (9) gives

$\begin{matrix}{T_{eq} = {T_{s} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{{\frac{0.00963\left( {P\; V} \right)^{0.5}}{\left\lbrack {0.5\left( {T_{s} + T_{a}} \right)} \right\rbrack^{0.04}L^{0.5}}\left( {T_{s} - T_{a}} \right)} +} \\{{{ɛ\sigma}\left( {T_{s}^{4} - T_{a}^{4}} \right)} + {\overset{.}{w}l_{e}} - {\alpha\; G} - {\frac{k}{s}\left( {T_{b} - T_{s}} \right)}}\end{bmatrix}} \right\}^{1/{({1 + \phi})}}}} & (12)\end{matrix}$which is a detailed expression for the equivalent temperature, T_(eq).If the surface flow had been turbulent, substitution of Eq. (11b) intoEq. (9) would have resulted in a similar equation for T_(eq). In eithercase, it is obvious that an inanimate object cannot “sense” the chillingeffect represented by this equivalent temperature. This temperature isnot a fictitious temperature but an actual temperature encountered bythe object in a cooling environment. Consequently, Eq. (12) becomesuseful in determining not only the magnitude of this cooling but alsoproviding guidelines as to what human actions might be taken in certainsituations to minimize the cooling. Such is the situation as describedin the following paragraphs along with two additional and potentiallyuseful applications of Eq. (12), one of which may already exist.

Consider a motor vehicle that must be parked outdoors overnight in acold winter environment. The owner is fully aware of the addeddifficulty of starting a very cold engine the following morning, andtherefore is likely to take some action to minimize the overnightcooling of the engine. In doing so, he is attempting to “maximize theengine's T_(eq)”. His first action is likely to be a covering of theengine, whereby any wind-generated forced convection heat loss in Eq.(12) is eliminated and the radiation heat loss is reduced if notcompletely eliminated. An alternative to covering the engine, but not aseffective, would be the parking of the vehicle on the leeward side of awind obstruction. In this instance, the convective heat loss would bereduced if not completely eliminated and the radiation heat loss wouldremain unchanged. As far as the other terms in Eq. (12) are concerned,the evaporation ({dot over (w)}l_(e)) and solar radiation (αG) termswill play no part, but the conductive heat term (k/s)(T_(b)−T_(s)) will.It can be viewed as residual engine heat remaining from the previousday's operation, which will partly counteract the heat losses due toforced convection and radiation, and thus aid in the subsequent startingof the engine. In this case where a conduction of heat takes place fromthe engine's interior to its surface, the actual value of φ is neitherknown nor easily determined. However, an estimate of the T_(eq) could bemade by assuming φ=0.25 and solving for T_(eq) from Eq. (12).Individuals experienced in winter living in the northern US latitudescan testify to the validity of these actions as it pertains to vehicleoperation.

The above paragraph clearly shows an application of the equivalenttemperature in Eq. (12) to a stationary vehicle. But Eq. (12) would alsoapply if in its development the segment in FIG. 1A was not stationarybut rather was moving with a velocity (V). Now if this moving segment isviewed as a vehicle in motion and if T_(eq) and T_(s) were measured onthe windward and leeward sides of the moving vehicle, respectively, andthe evaporative, solar and heat conduction values were known andaccounted for, it would be possible to determine the value of theambient temperature (T_(a)). This may be the approach used in presentday vehicles to measure the ambient temperature when the vehicle is inmotion.

Eq. (12) offers the real possibility of being used to predict the onsetof icing on aircraft surfaces, see e.g., Wilson, J. R., “Icing researchheats up”, Aerospace America, pp. 38-43, May Issue, 2006. This can beviewed as the counterpart to the onset of facial freezing as discussedin detail herein. Therefore, the analytical development that wasnecessary to arrive at a “time to freeze” as disclosed herein, could berepeated to determine a corresponding onset time for ice formation. Sucha development is possible because Eq. (12) realistically treats allmodes of heat transfer that contribute to the equivalent temperature ofthe wing and fuselage surfaces prior to takeoff (V=0) and in flight(V>0). The laminar forced convection term in Eq. (12) could be replacedby the turbulent forced convection term in the event high angle ofattack flight causes localized turbulence. In either convection term,the ambient pressure (P) would be expressed as a function of altitude(H) making Eq. (12) readily adaptable to all flight altitudes. Theradiation heat loss term of Eq. (12) would remain unchanged and would bethe only means of aircraft surface cooling prior to takeoff. Theevaporative heat loss term, q_(e), (see Eq. 3) would not exist while thesolar insolation heat gain term (αG) would correct for the absence orpresence of sunshine. A key term is the heat conduction term, k/s(T_(b)−T_(s)), which represents heat flow from inside the aircraftstructure to the outer surfaces. This heat flow would delay iceformation; after ice formation it would be the necessary heat flowrequired to loosen the ice such that it could be swept from the surfacesby windshear. In its final form Eq. (12) would calculate the equivalentsurface temperature (T_(eq)) as a function of flight conditions (H, V,T_(a)) and the initial surface temperature (T_(s)). This surfacetemperature will decrease due to the surface cooling during flight.Since it is desired to know the time when the value of T_(s) decreasesto the point that the water component in the air freezes on the surface,presumably at 32° F., an expression for T_(s) as a function of time (t)would have to be determined. Temperature distributions along surfaces inlaminar or turbulent flows are available in forms of analyticalsolutions with respect to wing de-icing. More accurate solutions may beobtained using computational fluid dynamics (CFD). This developmentwould parallel the development of a similar expression for the facialtemperature (T_(f)) as a function of time as developed in thisapplication. With the final expression T_(s) vs. t, the predicted onsetof ice formation would take place at the time (t) when T_(s)=32° F.

Wind Chill Temperature Equation: Eq. (12) is the basic equation for theequivalent temperature of a two-dimensional surface. The wind chillmodel developed herein essentially modifies this equation for theequivalent temperature (T_(eq)) to derive a corresponding equation forthe wind chill temperature (T_(wc)) as it applies to humans. It isnatural to assume that the human head can best be simulated by acylindrical surface. Osczevski correctly recognized that the realsensation of wind chill relates to cooling of the exposed face, seeOsczevski, R. J., “The basis of wind chill”, Arctic., Vol. 48, No. 4,pp. 372-382, 1995. Osczevski used a cylindrical face simulator in a windtunnel to obtain forced convection heat transfer coefficients. The Sipleand Passel experiment also used a cylinder. A comparison of Siple andPassel's results with those of Osczevski showed an agreement thatprompted Osczevski to observe that “[a]lthough the cylinder used bySiple and Passel was too small to represent a human body or even a head,it was nearly the perfect size to represent a face in the wind.” Theagreement was partly due to Siple and Passel's fortuitous choice ofcylinder size. Buettner had also shown some agreement between measuredand computed facial temperatures due to forced convection cooling of acylinder representing the human head, see Buettner, K., “Effects ofextreme heat and cold on human skin, I., Analysis of temperature changescaused by different kinds of heat application, II., Surface temperature,pain and heat conductivity in experiments with radiant heat”, J. Appl.Physiology, Vol. 3, No. 12, pp. 691-713, 1951. The Bluestein and Zecherwind chill model also assumed the face to be the surface of a cylinder.

The assumption that the human head can be viewed as a cylinder was alsomade by the inventors in the wind chill model of the present invention,and this assumption resulted in changes in the heat loss/gain terms ofEq. (12). There was a significant change in the forced convection term(q_(fc)) where the characteristic dimension (L) of the two-dimensionalsurface now becomes the vertical length or height of the cylinder. Thecoefficients in the radiation (q_(r)), evaporation (q_(e)), insolation(q_(i)) and conduction (q_(b)) terms take on special meanings, notbecause of the surface change, but because they now apply to humans. Thefollowing sections discuss each of these changes after defining thecylindrical equivalent of a human head.

The cylinder is viewed as being vertical with its longitudinal axisnormal to the wind. Its length (L) and diameter (D) must be specified.Because it can be demonstrated that an adult human head can be closelyapproximated by a 7 in. (17.78 cm) diameter cylinder that is 8.5 in.(21.59 cm) in length, the selected dimensions for a cylindrical model ofa human head were D=7 in. and L=8.5 in. These are essentially the samedimensions as used by Bluestein and Zecher in the development of theirwind chill model. The flat surface segment shown in FIGS. 1A-B in thedetermination of the equivalent temperature now becomes a segment offacial skin comprising the surface of the cylinder.

FIG. 2 is a diagram of a segment of a cylindrical facial surface used inthe determination of the wind chill temperature (T_(wc)), according tothe present invention. FIG. 2 illustrates the facial temperature (T_(f))on the exposed surface and all the attendant heat losses and gains. Theskin is composed of an outer layer called the epidermis, an adjacentinner layer called the dermis and an innermost layer called thesubcutaneous fat. Some difficulty was encountered in findingwell-defined values for the thickness of each layer. This may be partlydue to the fact that skin thickness depends on its location on the body.

A cross-sectional sketch of the skin drawn to scale, may be found inWorld Book Encyclopedia, Vol. 17, 404d, p. 405, 1978. The World BookEncyclopedia states that the thickness of the dermis varies from 3 mm(0.0098 ft) on an individual's back to 1.6 mm (0.0052 ft) on the eyelid.From this it was assumed that the thickness of the dermis on the facewas s_(d)=2.5 mm. Based upon the sketch, the corresponding thickness ofthe epidermis was s_(e)=0.87 mm and that of the subcutaneous fat wass_(sf)=1.96 mm. From these values, the cylindrical wall thicknesscorresponding to total skin thickness in FIG. 2 was determined to bes=5.33 mm.

The outer part of the epidermis consists of layers of lifeless fat cellsthat provide the body with a protective covering and a barrier toprevent loss of water through the skin. The inner part, at theepidermis-dermis interface, consists of live cells including nerve cellsthrough which the effects of the wind chill are sensed. Thedermis/subcutaneous fat interface is the location of the glands thatproduce the sweat in the evaporative heat loss.

It is important to note that the ratio of the facial skin height (L) of7 in. to the total skin thickness (s) of 5.33 mm (0.2098 in.), i.e.,L/s, is approximately 33. Since this skin thickness (s) is smallrelative to the facial skin height, it is reasonable to assume thatconduction through the skin occurs exclusively in one direction. Thereasonableness of this assumption and the accuracy of thisone-dimensional treatment will be verified below when comparing thefacial freezing time as calculated here with an infinite series solutionof the one-dimensional transient heat conduction in a plane wall.

Forced Convection Heat Loss (q_(fc)): The laminar forced convectioncoefficient (h_(fc, l, s)) for the two-dimensional segment shown in Eq.(11a) and used in Eq. (12) for the equivalent temperature, T_(eq), mustnow be replaced with the corresponding equation for a cylinder. At thispoint it should be noted that for a cylinder there is no need for anequivalent expression of the turbulent forced convection coefficient(h_(fc, t, s)) in Eq. (11b) since laminar flow will extendcircumferentially outward to about 80 degrees on either side of the windstagnation point. This laminar region is essentially the entire facialportion of the head that is directly exposed to the wind.

From Harms et al., the incompressible form of the laminar forcedconvection coefficient (h_(fc)) for this laminar stagnation region on acylinder may be expressed as:

$\begin{matrix}{{Nu} = {{h_{fc}\frac{x}{k}} = {1.14({Re})^{0.5}\left( \Pr \right)^{0.4}}}} & \left( {13a} \right)\end{matrix}$where the characteristic length (x) in the Re number is the cylinderdiameter (D). Equation (13a) is a special case of the general equationfor forced convection over a cylinder, Nu_(D)=h D/k=C(Re_(D))^(m)(Pr)^(n). Selecting the exponent m to be 0.5 reduces thegeneral relation to laminar flow. Furthermore, selecting the coefficientC to be 1.14 reduces the general relation to the laminar flow to thestagnation point. Thus, this special case applies to laminar stagnationheat transfer for all velocities. Neglecting the compressibility effectsand the usage of the reference temperatures (T^(l)) of Eq. (10c), asdone before, Eq. (13a) reduces to the following form for the laminarforced convection coefficient for a cylinder as used in this study:

$\begin{matrix}{h_{{fc},l,c} = \frac{0.03238\left( {P\; V} \right)^{0.5}}{\left\lbrack {0.5\left( {T_{s} + T_{a}} \right)} \right\rbrack^{0.04}D^{0.5}}} & \left( {13b} \right)\end{matrix}$where again the temperatures (T_(s) and T_(a)) are expressed in ° F.absolute (or ° R). Note the similarity of Eq. (13b) to that for thetwo-dimensional segment as shown in Eq. (11a).

Suppose the diameter (D) of a cylinder is equal to the length (L) of asegment. Then the ratio of the coefficients (h_(fc,l,c)/h_(fc,l,s)), is3.36, which shows that the forced convection cooling of the cylinderwith its longitudinal axis normal to the wind direction is 3.36 timesgreater than that for a two-dimensional surface aligned so as to beparallel to the wind. Perhaps this explains why a person facing into thewind on a cold winter day may instinctively turn his head to the side tolessen the cold sensation.

Eq. (13b), like Eq. (11a), clearly shows the dependency of the forcedcoefficient on the wind velocity (V) and the ambient pressure (P). Thisvelocity, V, must be that at head level, and must also be expressed inft/sec. However, in dealing with wind chill, it is more convenient toexpress velocity, V, in mph. Since the velocity at head level may undercertain situations be less than that at the National Weather Service(NWS) standard 10 m level, the concept of a wind reduction factor (WRF),as discussed more fully below, was introduced.

Making these changes, the velocity in ft/sec at head level becomes(1.467) (1−WRF) V where V is the velocity in mph at the NWS 10 m (32.81ft) level. The ambient pressure, P, is a function of altitude. Expressedin lbf/ft² in terms of its sea level value (2116.8 lbf/ft² (1 atm)) andaltitude (H) in ft using the correlation disclosed in John, J. E. A. andHaberman, W. L., Introduction to Fluid Mechanics, 2nd ed., PrenticeHall, pp. 24-26, 1980, the pressure is 2116.8 [1−(6.92×10⁻⁶)H]^(5.21).Putting these expressions for V and P into Eq. (13b), the final form ofthe forced convection coefficient for the cylindrical surface becomes:

$\begin{matrix}{h_{fc} = \frac{(1.8062)\left\{ {\left( {1 - {WRF}} \right){V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}} \right\}^{0.5}}{\left\lbrack {0.5\left( {T_{s} + T_{a}} \right)} \right\rbrack^{0.04}D^{0.5}}} & \left( {13c} \right)\end{matrix}$It can be shown that the absolute humidity is a function of ambientpressure, ambient temperature, and the relative humidity. The aboveexpression for the convective heat transfer coefficient is a function ofthe ambient pressure and ambient temperature, and therefore accounts forthe effects of humidity.

Radiation Heat Loss (q_(r)): The radiation heat loss (q_(r)) as shown inEq. (2) and used in Eq. (12) remains the same but is expressed in termsof a radiation coefficient (h_(r)) which applies to humans. Uponexpansion, Eq. (2) becomes:q _(r)=εσ(T _(s) +T _(a))(T _(s) ² +T _(a) ²)(T _(s) −T _(a))=h _(r)(T_(s) −T _(a))  (14a)where the radiation heat loss coefficient is:h _(r)=εσ(T _(s) +T _(a))(T _(s) ² +T _(a) ²)  (14b)The surface emissivity of the human head was determined to be ε=0.8based upon a dynamic model disclosed in Fiala, D., Lomas, K. J. andStohrer, M., “A computer model of human thermoregulation for a widerange of environmental conditions: the passive system”, J. Appl. Phys.,Vol. 98, pp. 1957-1972, 1999. The Fiala et al. model was developed forevaluating the human response to a cold, cool, neutral, warm, or hotenvironment. With this value of ε and with the Stefan-Boltzmann constantα=1.714×10⁻⁹ Btu/hr-ft²−° R⁴ (5.670×10⁻⁻⁸ W/m²-° K⁴), the radiation heatloss coefficient becomes:h _(r)=(1.3712×10⁻⁹)(T _(s) +T _(a))(T _(s) ² +T _(a) ²)  (14c)

Evaporation Heat Loss (q_(e)): The evaporation heat loss (q_(e)) fromthe skin segment as shown in Eq. (3) is the total heat loss resultingfrom the evaporation of sweat from the skin surface and from a naturaldiffusion of water through the skin layer. In this expression, {dot over(w)} is the total water evaporation flux rate and l_(e) is the latentheat of evaporation for water. It is known that l_(e)=539.49 cal/gm(971.08 Btu/lbm) so that q_(e)=971 {dot over (w)}, see, e.g., Eshbach,O. W., Handbook of Engineering Fundamentals, John Wiley & Sons, Inc.,pp. 1-142 and 7-119, 1952.

The problem here is the determination of this total water evaporationflux rate which is based upon the sweat rate and the diffusion rate.From the 1993 ASHRAE (American Society of Heating, Refrigerating and AirConditioning Engineers) Handbook, Fundamentals, sweating is shown to bea thermoregulatory process in which a risen body core temperature(T_(CR)) may be lowered to its equilibrium or neutral value (98.2° F. or36.78° C.) based upon warm signals from the core and skin. Sinceinformation was lacking in the calculation of both the sweat rate andthe diffusion rate, these values were obtained from Vogel, H. C. A., TheNature Doctor, Instant Improvement, Inc., p. 316, 1991. Vogel disclosesthat the diffusion rate varies from 1.7 to 2.6 pints/day (0.85 to 1.3quarts/day). This represents a heat loss from the skin that persists atall times in the absence of perspiration. Using an average value of 2.2pints/day (1.1 quarts/day), this value of the diffusion rate must beconsidered at all times in the calculation of wind chill.

Another source of water loss may be from sweat. Clayman, C. B., TheAmerican Medical Association, Home Medical Encyclopedia, Vol. 2, RandomHouse, p. 1070, 1989, discloses that the sweat rate is 1.1 pints/day(0.55 quarts/day) in a cold climate and 5.0 pints/day (2.5 quarts/day)in a warm climate. Clayman notes that these are average values and assuch represent cases where individuals are performing a minimum ofphysical activity. Based on these values, the total water loss rate,diffusion plus sweat, would be 3.3 pints/day (1.65 quarts/day) in a coldclimate during minimum physical activity. But the sweat rate dependsupon the severity of the physical activity. Vogel has stated that anindividual perspiring under extreme conditions in a steam or sauna bathor living in the tropics can experience a total (diffusion and sweating)water loss rate that is 10 times greater than the average diffusion rateof 2.2 pints/day (1.1 quarts/day). Because even the most extremewintertime activities are expected to result in a total water loss ratemuch less than this, a rather arbitrary assumption was made that thistotal water loss rate for maximum physical activity would be 5 times,not 10 times, the average diffusion rate. Thus, the total water lossrate from the skin surface for maximum physical activity would be5×2.2=11.0 pints/day (5.5 quarts/day).

Swerdlow, J. K., “Unmasking Skin”, National Geographic, pp. 36-63,November, 2002, discloses that the total skin area of an average personis 21 ft² (1.95 m²), which is close to values quoted by other sources.It was assumed that diffusion and sweating occur over the entire 21 ft²(1.95 m²). If it does not, then the resultant water evaporation fluxrate would be conservative. For minimum physical activity in a coldclimate where the total water loss rate is 3.3 pints/day (1.65quarts/day), the minimum evaporation flux rate is {dot over(w)}_(min)=0.00655 lbm/hr-ft² (0.032 kg/hr-ft²) and for maximum physicalactivity in a cold climate where the total water loss rate is 11.0pints/day (5.5 quarts/day), the maximum evaporation flux rate is {dotover (w)}_(max)=0.0283 lbm/hr-ft² (0.1066 kg/hr-ft²). The choice as towhich evaporation flux rate to use was determined by the degree ofstrenuousness of the individual's activity.

Table 1 below, shows values of the metabolic heat rate (M_(act))required for various activities and which will be used later todetermine the metabolic heat gain. The following assumptions were madein order to specify the evaporation flux rate for each differentactivity: (1) when M_(act)≦40 Btu/hr²-ft² (126.18 W/m²), the physicalactivity is at a minimum and {dot over (w)}=0.00655 lbm/hr-ft² (0.032kg/hr-ft²), and (2) when M_(act)>40 Btu/hr²-ft² (126.18 W/m²), thephysical activity is at a maximum and {dot over (w)}=0.02183 lbm/hr-ft²(0.1066 kg/hr-ft²). With the evaporation flux rate ({dot over (w)})known, along with the known value of the latent heat of evaporation(l_(e)) for water, the evaporation heat loss, q_(e)=(971){dot over (w)}can be calculated. This value of q_(b) provides an estimate of theevaporation heat loss.

TABLE 1 Metabolic heat generation M_(act) for various activitiesM_(act,) Btu/hr-ft² Activity (W/m²) Standing relaxed 22 (69.40) Walkingabout 31 (97.79) Walking on level surface 2 mph (3.22 Km/hr)  37(116.72) 3 mph (4.83 Km/hr)  48 (151.42) 4 mph (6.44 Km/hr)  70 (220.82)Driving Snowmobile 18 to 37  (56.8 to 116.72) Heavy vehicle  59 (186.12)Pick and shovel work 74 to 88 (233.44 to 277.60) Skiing Cross country 83 (261.83)* Downhill  96 (302.84)* *Estimated by the inventors

Solar Heat Gain (q_(i)): The solar heat (q_(i)) received by the skinsegment as shown in Eq. (4) depends on the insolation value (G). Ifsunshine is present, G>0, in darkness G=0. In the present case, wherethe major concern is an excessive wind chill leading to possible facialfreezing, insolation or solar radiation heating of the face has amoderating effect. Because the heating of the face is maximized on aclear day when the sun's rays are perpendicular to the facial surface,this maximum value depends on the latitude of the individual's location.The value also depends on the season of the year. Consider, for example,the case of an individual at a northern latitude angle (LAT). In thesummer, when the sun is high in the sky the angle of incidence of thesun's rays upon an upright individual's face may be small, resulting inrelatively little solar heating of the face. Contrast this with thewinter season when the sun is low in the sky and the sun's rays may benearly perpendicular to the face, thus maximizing the facial heating. Itis fortunate that this radiation heating is at its maximum in the winterseason when its benefit can best be utilized. Finally, the insolationvalue at a given location is dependent upon the daily number of hours ofsunshine at the location.

New England Solar Electric Inc., The Solar Electric Independent HomeBook, New England Solar Electric Inc., Chap. 8, pp. A18-A39, 1998,discloses insolation data by the month for 221 cities across the 50states as compiled by the U.S. Department of Energy. The data in thisreference was used to devise a method for selecting the correct value ofinsolation to be used in the calculation of wind chill. This was done bydividing the portion of the lower 48 states that would most likely besubjected to wind chill into three latitude (LAT) regions of 50 each.These regions are 35°<LAT<40°, 40°<LAT<45° and 45°<LAT<50°, where LAT isthe latitude angle of a particular location within a region. Thecombined span of these three regions may be visualized by noting thatthe 35° latitude line follows the southern boundary of Tennessee whilethe 50° latitude line lies close to the southern boundary of theCanadian provinces. Within each region, the average monthly insolation(G_(av)˜KW-hr/m²) was determined for four or more cities over a fourmonth period from November through February. During this four monthperiod, the nearly normal angle of incidence of the sun's rays on acollecting surface, in this case, an individual's head, was LAT+15°.

The values of G_(av) are those corresponding to this angle of incidence.The values of G_(av) were then used to obtain the average values(G_(LAT)) over each latitude region. These values are shown in Table 2,below, where G_(LAT) has also been expressed as G in English units(Btu/hr-ft²) for use in this model. It should be noted that the valuesfor G (37.78 and 42.66 Btu/hr-ft² (119.17 and 134.58 W/m²)) in the twonorthern regions, where severe wind chill and facial freezing is apossibility, are only 12 to 13% of the value of 240 cal/s-m² (318.53Btu/hr-ft²) used by Steadman (1971) for an upright human after hecorrected his direct insolation value for a low angle solar altitude.

TABLE 2 Insolation values for latitude regions in the lower 48 states GLatitude G_(av) G_(LAT) W/m² LAT City KW-hr/m² KW-hr/m² (Btu/hr-ft²)45°-50° Bangor, ME 2.76 2.86 119.17 Minneapolis, MN 2.81 (37.78)Bismarck, Fargo, ND 2.75 Miles City, Great 3.12 Falls, MT 40°-45°Boston, MA 2.62 3.23 134.58 Chicago, IL 2.81 (42.66) Huron, Pierre, SD3.26 Casper, Rock Springs, 4.21 Sheridan, WY 35°-40° Raleigh,Greensboro, NC 3.70 4.43 184.58 Kansas City, Springfield, 3.50 (58.51)St. Louis, MO Colorado Springs, 4.99 Denver, Grand Junction, COPrescott, AZ 5.53

The thermal absorptivity (α) of the human skin was determined fromresults of a study disclosed in Buettner, K., The effects of naturalsunlight on human skin, Proceedings of the first InternationalConference, Sponsored jointly by the Skin and Cancer Hospital, TempleUniversity Health Sciences Center and the International Society ofBiometeorology, Pergamon Press, 237-249, 1969. In this study, thereflectivity of sunshine from white skin was found to be 35%. Assumingno transmissivity, the absorptivity was 65% and therefore the thermalabsorption coefficient (α) was 0.65. For dark skin, the absorptivitywould be expected to be greater than 0.65; however, since this value wasnot known an α=0.65 was used in this model. With G determined, the solarheat gain is q_(e)=(0.65) G.

Metabolic Heat Gain (q_(b)): What was previously described as heatconduction (q_(b)) into the segment in Eq. (5) now becomes the metabolicheat flow from deep inside the body, called the core, into themulti-layered skin segment. This is more clearly understood in terms ofthe two-component model disclosed in Gagge, A. P., Stolwijk, J. A. J,and Nishi, Y., “An effective temperature scale based on a simple modelof human physiological regulatory response”, ASHRAE Transactions, Vol.77, No. 1, p. 247, 1971, in which the human body is represented as twoconcentric cylinders. Described in the 1993 ASHRAE Handbook, the innercylinder represents the body core (skeleton, muscle, internal organs)and the outer annular cylinder, the skin layer. The metabolic heat (M)produced within the core of the body is the heat required for anindividual's activity (M_(act)) plus that required for shivering(M_(shv)), should that occur. Typical values of M_(act) for variousactivities were obtained from the 1993 ASHRAE Handbook and OSHATechnical manual, Occupational Safety and Health Administration, Heatstress, Section III, U.S. Department of Labor, Chap. 4, 2003, and areshown in Table 1, above. Shivering is a thermoregulatory process throughwhich the core temperature (T_(CR)) may be restored to its equilibriumor neutral value of 98.2° F. after a lowering. Although (M_(shv)) can betheoretically determined based on cold signals from the core and theskin, it was not considered here because of the uncertainty of itsoccurrence and its anticipated minimal effect on the final wind chilltemperature.

If a portion of the heat produced within the core is expended as work(W) performed by the muscles, then the net heat production (M_(act)−W)is dissipated to the environment through the skin layer or is stored inthe core causing the core temperature (T_(CR)) to rise. The possiblerise in core temperature was not considered because most wintertimeactivities are not so extreme that the metabolic heat cannot becompletely dissipated through the skin. Therefore, the core temperaturewas assumed to remain constant. Calculations using large values ofM_(act) verify this. In addition, work performed by the muscles cannotbe specifically defined for the general populace when predicting windchill temperatures; consequently, it too was neglected (W=0). Based onthe two-cylinder concept of the human body, the metabolic heat flow(q_(b)) becomes a conductive heat transfer of heat from the corecylinder at a constant temperature (T_(CR)) to the skin cylinder whoseouter surface is at the facial temperature (T_(f)). In FIG. 2, the skinsegment which is a section of the outer skin cylinder is convenientlyshown as a flat surface where the left side is the interface between thecylinders across which the metabolic heat transfer takes place. Theexpression for this heat transfer is:q _(b) =K(T _(CR) −T _(f))  (15)This expression is similar to Eq. (5) except that the k/s term isreplaced by a more complex conductance term (K). The 1993 ASHRAEHandbook states that this heat is transferred from the core to the skinby conductance (κ) at the interface between them and by convection dueto the skin blood flux rate ({dot over (m)}_(b,sk)) within the skinsegment. Therefore the complex conductance term becomes:K=(κ+C _(p,b) {dot over (m)} _(b,sk))with κ specified as 0.93 Btu/hr-ft²-° F. (5.28 W/m²-° C.) and with theblood specific heat C_(p,b)=1.0 Btu/lbm-° F. (4.186 KJ/kg-° C.). Theconductance term becomes:

$\begin{matrix}{K = {\left( {{.93} + {C_{p,b}{\overset{.}{m}}_{b,{sk}}}} \right)\frac{Btu}{{hr}\text{-}{ft}^{2}\text{-}{^\circ}\mspace{14mu}{F.}}}} & \left( {16b} \right)\end{matrix}$and so it is directly proportional to the skin blood flux rate which islimited to the range, 0.1 lbm/hr-ft²<{dot over (m)}_(b,sk)<18 lbm/hr-ft²(0.49 kg/hr-m²<{dot over (m)}_(b,sk)<87.88 kg/hr-m²). Increasingactivity (M_(act)), increases the skin blood flux rate and thusincreases the metabolic heat flow from the core to the skin segment.

The above conductance term (K) is defined differently in athermoregulatory model developed by Havenith in which he determined thehuman response to heat and cold exposure, see Havenith, G.,“Individualized model of human thermoregulation for the simulation ofheat stress response”, J. Appl. Physiology, Vol. 90, pp. 1943-1954,2001. Havenith also used the two cylinder model developed by Gagge etal., but rather than defining the interface as in FIG. 2, Havenithdefined the interface such that the outer cylinder, called the shell,consisted of muscle, fat, and skin, where fat refers to the subcutaneousfat layer and skin refers to the epidermis and dermis layers. Theresistance to the heat transport from the core to the shell is thereforedue to the combined resistance of the muscle, fat, and skin. When theblood vessels in the shell are constricted, heat transmission throughthe shell is low and mainly by conduction. This situation is typified bya state of low activity (low M_(act)). Increasing activity (higherM_(act)) increases the blood flux rate within the muscle and skin of theshell, thereby adding a convective component to the heat transfer. Asalready stated, the result is that the greater the activity the greaterthe heat flow to the skin, so that during wintertime the moderatingeffect on the wind chill temperature as sensed by the individual becomesgreater.

Havenith determined the thermal resistance (R) of the shell, that is theinverse of the thermal conductance (1/K), by adding in series theresistance due to the skin blood flux rate (R_(b)), the resistance dueto the muscle insulation and the muscle blood flux rate (R_(m)) and theresistance of the fat layer and two skin layers (R_(sk)) to get thefollowing equation:K=R ⁻¹=(R _(b) +R _(m) +R _(sk))⁻¹  (17a)where each of the three resistances is expressed in SI units (m²-°C./W). The resistance due to blood flux rate is:

$\begin{matrix}{R_{b} = \frac{1}{\eta\; C_{p,b}{\overset{.}{m}}_{b,{sk}}}} & \left( {17b} \right)\end{matrix}$where Havenith refers to η as a countercurrent heat exchange efficiency.This efficiency (η) is a measure of the arterial blood's ability tocarry heat to the skin segment. The greater the value of η, the smallerthe values of R_(b) and R, the greater the conductance (K) and thegreater the heat flow to the skin. The blood vessels in the skin areboth arterial and venous in nature. The arteries carry heated blood fromthe core to the inner part of the epidermis but not into its outer part,which consists of multiple layers of dead cells. The veins carry thecooled returning blood to the core. The flow in the veins is counter tothat in the arteries, and as a result, some of the heat in the incomingarterial flow directed toward the epidermis is absorbed by the outgoingvenous flow leaving the epidermis. This heat exchange mechanism is thesame principle employed in counterflow heat exchanges where theirefficiency, η_(cf), is a measure of the ability of the colder flow toremove heat from the warmer. Consequently, the greater the heat removalefficiency (η_(cf)) of the veins, the lesser the heat transportefficiency (η) of the arteries; therefore η can be expressed asη=1−η_(cf). Havenith assumed η=0.5 and then calculated the specific heatof the blood (C_(p, b)) on the basis of what is called a “standard man”whose fat content is 15% of his total mass. Knowing the specific heatfor fat as 2.51 J/gm-° C. (0.6 Btu/lbm-° F.) and that for other bodytissue (skin, skeleton, muscles) as 3.65 J/gm-° C. (0.87 Btu/lbm-° F.),C_(p,b)=(0.15) (2.51)+(1−0.15)(3.65)=3.48 J/gm-° C. or 3480 J/kg-° C.(0.83 Btu/lbm-° F.). With these values, Eq. (17b) becomes:

$\begin{matrix}{R_{b} = \frac{1}{1740{\overset{.}{m}}_{b,{sk}}}} & \left( {17c} \right)\end{matrix}$where the blood flux rate ({dot over (m)}_(b,sk)) is expressed inkg/sec-m².

The resistance due to the muscle may be expressed as:

$\begin{matrix}{R_{m} = \frac{.05}{1 + \left( \frac{M_{act} - 65}{130} \right)}} & \left( {17d} \right)\end{matrix}$where 0.05 is called the maximal muscle insulation and the denominatorrelates muscle blood flux rate to energy consumption through themetabolic heat rate (M_(act)) expressed in W/m² for a given activity.

The resistance due to the subcutaneous fat layer and the two otherlayers (dermis, epidermis) of the skin is:R _(sk)=0.0048(s−2)+0.0044  (17e)where s is the total thickness of all three layers expressed in mm. Withs=5.33 mm (0.0175 ft) from FIG. 2, Eq. (17e) becomes:R_(sk)=0.02038  (17f)Substituting Eqs. (17c), (17d) and (17f) into Eq. (17a) the thermalresistance (R) of the core is:

$\begin{matrix}{R = {\frac{1}{1740{\overset{.}{m}}_{b,{sk}}} + \frac{.05}{1 + \left( \frac{M_{act} - 65}{130} \right)} + {.02038}}} & \left( {17g} \right)\end{matrix}$Expressing the blood flux rate ({dot over (m)}_(b,sk)) in kg/hr-m²rather than kg/sec-m² and then expressing the conductance (K) as theinverse of Eq. (17g), K becomes:

$\begin{matrix}\begin{matrix}{K = R^{- 1}} \\{= {\left\lbrack {\frac{2.069}{{\overset{.}{m}}_{b,{sk}}} + \frac{1}{10 + {{.1538}\mspace{11mu} M_{act}}} + {.02038}} \right\rbrack^{- 1}\left( \frac{W}{m^{2}\text{-}{^\circ}\mspace{14mu}{C.}} \right)}}\end{matrix} & \left( {17h} \right)\end{matrix}$Aside from the difference in units, this conductance term by Havenith isquite different from the conductance term of Eq. (16b) as obtained from1993 ASHRAE Handbook, Fundamentals. Each is a function of the skin bloodflux rate, which in turn is dependent upon the metabolic heat (M_(act))required by the activity. The following paragraphs describe theprocedure used to determine which of these expressions for K would beused in the development of this model.

It has already been stated that the core temperature (T_(CR)) would notrise but remain constant, since all of the metabolic heat (q_(b))flowing to the skin could be completely dissipated through the skin suchthat q_(b)=M_(act). Then from Eq. (15),q _(b) =K(T _(CR) −T _(f))=KΔT=M _(act)  (17i)With this equation, it was possible to determine the conductance (K)term and hence the blood flux rate ({dot over (m)}_(b,sk)) thatsatisfies each value of ΔT and M_(act). This required establishing arange of values for ΔT and M_(act). With T_(CR)=98.2° F. (36.78° C.) andwith an initial facial temperature T_(f)=91.4° F. (33° C.), the initialand minimum temperature differential is ΔT=98.2° F.−91.4° F.=6.8° F.(3.78° C.) at the moment of exposure to the wind. Assume, duringcontinued exposure, the face was allowed to cool down to 40° F. (4.44°C.). The final and maximum temperature differential would be ΔT=98.2°F.−40° F.=58.2° F. (32.33° C.). Thus, the temperature differential rangeis 6.8° F.<ΔT<58.2° F. (3.78° C.<ΔT<32.33° C.).

Over this range of ΔT, individuals may be engaged in various physicalactivities requiring differing amounts of metabolic heat (M_(act)). Atthis point, the range of M_(act) was assumed to be that which existedduring the Adamenko and Khairullin (1972) experiment, see Adamenko, V.N. and Khairullin, K. Sh., “Evaluation of conditions under whichunprotected parts of the human body may freeze in urban air duringwinter”, Boundary-Layer Meteorology, Vol. 2, pp. 510-518, 1972. In thisexperiment, the facial components (cheeks, nose, ears) of 40 people wereinstrumented to record the component temperature while the people wereengaged in different levels of activity while exposed to ambienttemperatures ranging from 10° C. to −40° C. (50° F. to −40° F.) andwinds up to 15 m/s (33.55 mph). Their estimate of heat production by thehuman body, that is M_(act), for all 40 people involved in theexperiment ranged from 0.08 cal-cm⁻²-min⁻¹ (17.7 Btu/hr-ft²) to 0.60cal-cm⁻²-min⁻¹ (132.7 Btu/hr-ft²). Therefore, the range of M_(act)considered here was 17.7 Btu/hr-ft²<M_(act)<132.7 Btu/hr-ft² (55.84W/m²<M_(act)<418.62 W/m²).

Determining which expression for K to use in this model went beyonddetermining K itself for a given ΔT and M_(act) from Eq. (17i); ratherit depended upon the magnitude of the blood flux rate {dot over(m)}_(b,sk), which is characteristic of this equation as well as Eq.(16b). Consider first the ASHRAE expression for K in Eq. (16b).Substituting it into Eq. (17i) and solving for {dot over (m)}_(b,sk) inunits of lbm/hr-ft² gives:

$\begin{matrix}{{\overset{.}{m}}_{b,{sk}} = {\frac{1}{C_{p,b}}\left\lbrack {\left( {{M_{act}/\Delta}\; T} \right) - {.93}} \right\rbrack}} & \left( {17j} \right)\end{matrix}$At the moment of exposure when ΔT=6.8° F. (3.78° C.), the activity levelwould likely be at the lowest level, that is M_(act)=17.7 Btu/hr-ft²(55.84 W/m²). With these values, the blood flux rate from Eq. (17j) is{dot over (m)}_(b,sk)=1.67 lbm/hr-ft² (8.15 kg/hr-m²). Now afterextended exposure, when ΔT=58.2° F. (32.33° C.), the activity level islikely to be at its maximum value of M_(act)=132.7 Btu/hr-ft² (418.62W/m²) and the corresponding blood flux rate is {dot over(m)}_(b,sk)=1.35 lbm/hr-ft² (6.59 kg/hr-m²). Thus the blood flux ratefrom the ASHRAE expression for K decreases as the activity levelincreases.

Determining the corresponding blood flux rates for the Havenithexpression for K in Eq. (17h) required an iterative procedure, sincethis K is a function of both {dot over (m)}_(b,sk) and M_(act). For theminimum values of ΔT and M_(act), the blood flux rate was found to be{dot over (m)}_(b,sk)=41 kg/hr-m² (8.4 lbm/hr-ft²). For the maximumvalue of ΔT and M_(act), the blood flux rate was {dot over(m)}_(b,sk)=43 kg/hr-m² (8.81 lbm/hr-ft²). The Havenith values differfrom the ASHRAE values in two respects. First, the larger blood fluxrate for the Havenith K occurs at the higher activity level, whereas forthe ASHRAE K it occurs at the lower activity level. Secondly, theHavenith blood flux rates are much larger than the ASHRAE values. Thesemuch larger values of blood flux rates may be the result of the shellconcept employed by Havenith. The shell consists of muscle in additionto the three skin layers. Equation (17c) relates primarily to theresistance of the skin blood flux rate. Equation (17h) relates to skinblood flux rate plus muscle resistance and muscle blood flux rate. Fromthis it is clear that the Havenith values include both the skin andmuscle blood flux rates. This may explain why the Havenith values of{dot over (m)}_(b,sk) are so much larger than the ASHRAE values.Furthermore, Havenith has stated that the skin blood flux rate decreaseswith increasing M_(act), as already demonstrated by the ASHRAE result,while the muscle blood flux rate increases with increasing M_(act).Intuitively, it might be reasoned that the muscle blood flux rate couldbe much greater than that in the skin. If that is so, then as M_(act)increases, the decrease in the skin blood flux rate is more than offsetby the increase in the muscle blood flux rate. The net effect is thatthe larger total blood flux rate (8.81 lbm/hr-ft² (43 kg/hr-m²)) wouldindeed occur at the higher values of M_(act) as demonstrated.

The above discussion suggests that either expression for K could be usedto calculate the metabolic heat flow. However, if the above assumptionthat the total blood flux rate in the Havenith approach is that of boththe skin and the muscle, then the use of Havenith's expression might bemore appropriate. The reason for this is that the metabolic heat rate(M_(act)) now determines the blood flux rate in the muscle as per Eq.(17d), and therefore its effect on muscle resistance (R_(m)) andconsequently its contribution to the value of K. This aspect of core toskin heat transfer is not present in the ASHRAE conductance term.Nevertheless, the average value of the skin blood flux rate ({dot over(m)}_(b,sk)) of 1.5 lbm/hr-ft² (7.32 kg/hr-m²) as determined in theASHRAE calculations was used in the evaluation of the Havenithexpression for K from Eq. (17i).

The selection of the more appropriate expression for K was determined inanother way, although in an unusual manner. Consider the two averageblood flux rates of 1.5 lbm/hr-ft² (7.32 kg/hr-m²) for the skin and 8.61lbm/hr-ft² (42.04 kg/hr-m²) for the muscle and skin. From the differenceof these two values, the muscle blood flux rate alone becomes 7.1lbm/hr-ft² (34.67 kg/hr-m²). With a blood density of 62.4 lbm/ft³(999.52 kg/m³), these two blood flux rates (1.5, 7.1) correspond toblood flux rate velocities of 0.29 and 1.37 in./hr (0.74 to 3.48 cm/hr).Based on one of the inventors' personal experience with influenzainoculation, pain accompanied with shivering was felt in the center ofthe underarm 5 to 6 hours after inoculation. This is defined as thestart of the body reaction time to the influenza strain injected intothe arm. The distance between the syringe's insertion point in the armand the center of the underarm was approximately 6.5 in. (16.51 cm).With this approximate distance, the skin blood velocity (0.29 in./hr(0.74 cm/hr)) and the muscle blood velocity (1.37 in./hr (3.48 cm/hr))correspond to approximate body reaction times of 22.4 and 4.7 hours,respectively. Assuming no other effects that might dramatically alterthis body reaction time, the time of 4.7 hours appears realistic sinceit is very close to the 5 to 6 hour reaction time experienced.Consequently, it lends credence to the validity of Havenith's value of Kand for that reason this K was selected in the calculation of q_(b).

Equation (17h) shows K to be dependent upon {dot over (m)}_(b,sk) andM_(act), both expressed in SI units. As already stated, the average {dotover (m)}_(b,sk) of 1.5 lbm/hr-ft² (7.32 kg/hr-m²) obtained from theASHRAE calculations was used in this expression for K after convertingit to units of 7.32 kg/hr-m². Although {dot over (m)}_(b,sk) takes onthis constant value, M_(act) depends on the individual's activity andmust be specified from Table 1, above, where M_(act) is listed forvarious wintertime activities. Listed in units of Btu/hr-ft², it isconvenient to maintain M_(act) in these units when substituting theminto Eq. (17h). To do so, the values of M_(act) in Table 1 must bemultiplied by 3.1546 to convert them to W/m². Making these substitutionsin Eq. (17h) and then converting the entire equation to English units,the equation for K becomes:

$\begin{matrix}{{K = {{\left( \frac{1}{5.6784} \right)\left\lbrack {{.2827} + \frac{1}{10 + {{.1538}(3.1546)M_{act}}} + {.02038}} \right\rbrack}^{- 1}\mspace{14mu}\left( \frac{Btu}{{hr}\text{-}{ft}^{2}\text{-}{^\circ}\mspace{14mu}{F.}} \right)}}{{{or}\mspace{14mu}{upon}\mspace{14mu}{simplification}}\;,}} & \left( {17k} \right) \\{K = {\left\{ {1.721 + \left\lbrack {1.7611 + {{.0854}\left( M_{act} \right)}} \right\rbrack^{- 1}} \right\}^{- 1}\left( \frac{Btu}{{hr}\text{-}{ft}^{2}\text{-}{^\circ}\mspace{14mu}{F.}} \right)}} & \left( {17l} \right)\end{matrix}$This expression for K is now used in Eq. (15) to calculate the metabolicheat transfer (q_(b)).

The above defined expressions for h_(fc) and h_(r) from Eqs. (13c) and(14c), the expressions for quantities q_(e) and q_(i), along with q_(b)from Eq. (15) and its conductance term from Eq. (17l) are thereplacements for their counterparts in Eq. (12) when the latter isapplied to humans. The surface temperature (T_(s)) in the forcedconvection, radiation, and metabolic heat terms is replaced with thefacial temperature (T_(f)) and the equivalent temperature (T_(eq)) nowbecomes the wind chill temperature (T_(wc)). The above quantities andEq. (8a) relating to the natural convection coefficient were substitutedinto Eq. (6) and the following equation for the wind chill temperature(T_(wc)) was obtained in the same manner as Eq. (9) for the equivalenttemperature (T_(eq)),

$\begin{matrix}{T_{wc} = {T_{f} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{\frac{(1.8062)\left\{ {\left( {1 - {WRF}} \right){V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}} \right\}^{0.5}}{{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.04}D^{0.5}} +}\left( {T_{f} + T_{a}} \right)} \\{\left( {1.3712 \times 10^{- 9}} \right)\left( {T_{f} + T_{a}} \right)\left( {T_{f}^{2} + T_{a}^{2}} \right)} \\{\left( {T_{f} - T_{a}} \right) + {(971)\overset{.}{w}} - {({.65})G} - \left\{ {1.721 +} \right.} \\{\left. \left\lbrack {1.7611 + {{.0854}\left( M_{act} \right)}} \right\rbrack^{- 1} \right\}^{- 1}\left( {T_{CR} - T_{f}} \right)}\end{bmatrix}} \right\}^{1/{({1 + \phi})}}}} & \left( {18a} \right)\end{matrix}$where in this study, the dimensions D=7 in. (0.5833 ft, 17.78 cm) andL=8.5 in. (0.7083 ft, 0.2159 m) must be expressed in ft, and where alltemperatures (T_(wc), T_(f), T_(a), T_(CR)) are expressed in ° F.absolute (or ° R) and where C₁ is determined from Eq. (8c).

Equation (18a) is the final form of the wind chill temperature. At thispoint, exponent φ is unknown. As already stated, for natural convectionfrom a heated vertical surface, plane or cylinder, φ=0.25 if the solesource of the heat is within the surface itself. This is not the casefor a human where the facial surface heat is being supplemented by themetabolic heat from the body core. In this case, φ will take on a valuedifferent from 0.25. Natural heat convection (q_(nc)) will occur in a nowind (V=0 mph (0 Km/hr)) environment when the facial surface heats theadjacent air layer, causing upward convection. In the absence of forcedconvection (q_(fc)) heating, the air layer will be heated by radiation(q_(r)) from the facial surface, which is not only being heated by themetabolic heat (q_(b)) but also by solar radiation (q_(i)). Theevaporative heat loss (q_(e)) will have a cooling effect on the adjacentair. Therefore q_(nc)=q_(r)+q_(b)+q_(i)−q_(e). With the components ofq_(nc) as shown in Eq. (18a) and with Eq. (8a) substituted in Eq. (7),where T_(s) and T_(eq) have been replaced by T_(f) and T_(wc), Eq. (7)becomes,(C ₁)L ^(−φ)(T _(f) −T _(wc))^((1+φ))=(1.3712×10⁻⁹)(T _(f) +T _(a))(T_(f) ² +T _(a) ²)(T _(f) −T _(a))+{1.721+[1.7611+0.0854(M_(act))]⁻¹}⁻¹(T _(CR) −T _(f))+(0.65)G−(971){dot over (w)}  (18b)which upon solving for T_(wc) becomes,

$\begin{matrix}{{T_{wc}\left( {V = 0} \right)} = {T_{f} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{\left( {1.3712 \times 10^{- 9}} \right)\left( {T_{f} + T_{a}} \right)\left( {T_{f}^{2} + T_{a}^{2}} \right)} \\{\left( {T_{f} - T_{a}} \right) + \left\{ {1.721 + \left\lbrack {1.7611 + {{.0854}\left( M_{act} \right)}} \right\rbrack^{- 1}} \right\}^{- 1}} \\{\left( {T_{CR} - T_{f}} \right) + {({.65})G} - {(971)\overset{.}{w}}}\end{bmatrix}} \right\}^{1/{({1 + \phi})}}}} & \left( {18c} \right)\end{matrix}$For specified values of T_(a), M_(act) and G, Eq. (18a) with V=0 and Eq.(18c) are two equations in two unknowns (T_(wc), φ). To simplify thesolution of these quantities, sunshine was assumed to be absent (G=0).This absence means that the values of φ so determined will reflect thelowest values of T_(wc) whenever Eq. (18a) is applied even if the effectof sunshine is subsequently included. In addition, by assuming theindividual's usual physical activity as “walking about”, then from Table2 M_(act)=31 Btu/hr-ft² (97.79 W/m²) and {dot over (w)}=0.00655lbm/hr-ft² (0.032 kg/hr-m²). By making these assumptions and noting thatT_(f)=91.4° F. (551.09° R) and T_(CR)=98.2° F. (557.89° R), values ofT_(wc) and φ from Eqs. (18a) and (18c) were determined as functions ofT_(a), the only remaining variable. A plot of φ as a function of T_(a)was curve fitted using TableCurve 2D to give the following expressionfor φ:φ=a+bexp[−(T _(a) /C)]  (18d)where a=0.46259934, b=0.077254543 and c=−59.573525, and where thecorrelation coefficient is r²=0.998884458. The wind chill temperature(T_(wc)) at a given ambient temperature (T_(a)) can now be determinedfrom Eq. (18a) after determining φ from Eq. (18d).

Equation (18a) can be viewed as a complete expression for the wind chilltemperature, since it considers all heat losses and gains thatcontribute to it. It is applicable to any individual engaged in one of avariety of physical activities from merely standing to being engaged ina strenuous activity. Knowing the specific location of the individual,the solar radiation (G) is known as well as the altitude (H). Thus, thewind reduction factor (WRF) can be determined as described below.

Equation (18a) is a detailed expression for the wind chill temperatures(T_(wc)) on the facial surface that reveals certain features of thistemperature that might not be immediately apparent. These featuresbecome evident when one considers the particular situation where thereis no sunshine (G=0), no wind reduction (WRF=0), and where there isnegligible heat conduction (K=0) and evaporation (w=0). These featuresare as follows:

-   -   (1) For all values of V, if T_(a)=T_(f) then T_(wc)=T_(a)=T_(f).        This means that if the facial surface temperature and the        ambient temperature are the same, a wind chill temperature does        not exist. Since this is true for V=0 as well, there is no        longer a need to define a calm condition as 4 mph (1.79 m/s)        when stating wind chill temperature in the absence of wind.    -   (2) For values of V>0, if T_(a)>T_(f), then T_(wc)>T_(f). This        means that when the ambient temperature is greater than the        facial temperature, what was a wind chill temperature now        becomes a wind warming temperature. The net affect is a warming        of the face rather than a cooling.    -   (3) A decrease in the ambient pressure as a result of an        increase in altitude (H) moderates or increases T_(wc).

Facial Temperature Equation: The wind chill model disclosed hereinpredicts the wind chill temperature immediately upon exposure and thetime to freeze if exposure continues. The determination of this time tofreeze required the development of a time dependent facial temperature(T_(f)) equation for the outer surface of the facial skin segment asshown in FIG. 2. This was accomplished using two different approaches.

(1) Lumped Capacitance Approach: In the lumped capacitance approach, allheat losses (q_(fc), q_(r), q_(e)) from the segment and the heat gains(q_(b), q_(i)) to the segment are lumped together in a determination ofthe facial temperature. This temperature variation was determined fromthe time variation of the average skin segment temperature (T_(SG))defined as T_(SG)=(T_(CR)+T_(f))/2 in FIG. 2. From the conservation ofenergy, the rate of change of the skin segment internal energy (Ė_(SG))is equal to the rate of change of the entering energy (Ė_(IN)) minus therate of change of the leaving energy (Ė_(OUT)), that is,Ė _(SG) =Ė _(IN) −Ė _(OUT)  (19)With A as the surface area of the skin segment, using the heat loss andheat gain terms from Eq. (18a) in Eq. (19) gives,

$\begin{matrix}{{{{{\rho C}_{p}{sA}} = {\frac{\mathbb{d}T_{SG}}{\mathbb{d}t} = {{\left\lbrack {{K\left( {T_{CR} - T_{f}} \right)} + {({.65})G}} \right\rbrack A} -}}}\mspace{11mu}\left\lbrack {\frac{(1.8062)\left\{ {\left( {1 - {WRF}} \right){V\left\lbrack {1 - \mspace{11mu}{\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}} \right\}^{0.5}}{\begin{matrix}{{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.05}D^{0.5}} + \left( {1.3712 \times 10^{- 0}} \right)} \\{{\left( {T_{f} + T_{a}} \right)\left( {T_{f}^{2} + T_{a}^{2}} \right)\left( {T_{f} - T_{a}} \right)} + {(971)\overset{.}{w}}}\end{matrix}}\left( {T_{f} - T_{a}} \right)} \right\rbrack}A} & (20)\end{matrix}$where ρ is the skin density, C_(p) is the skin specific heat and theskin segment thickness s=0.0175 ft (5.33 mm), as previously determined.Rationalizing Eq. (20) and noting that T_(CR) remains constant as thesegment cools, then,

$\begin{matrix}{{\frac{\mathbb{d}T_{SG}}{\mathbb{d}t} = {{\frac{1}{2}\left( {\frac{\mathbb{d}T_{CR}}{\mathbb{d}t} + \frac{\mathbb{d}T_{f}}{\mathbb{d}t}} \right)} = {\frac{1}{2}\frac{\mathbb{d}T_{f}}{\mathbb{d}t}}}}{{{and}\mspace{14mu}{{Eq}.\mspace{14mu}(20)}\mspace{20mu}{becomes}},}} & \left( {21a} \right) \\{\frac{\mathbb{d}T_{f}}{\mathbb{d}t} = {{\frac{2}{\rho\; C_{p}s}\mspace{45mu}\begin{bmatrix}{{K\left( {T_{CR} - T_{f}} \right)} + {({.65})G}} \\{{- \frac{\begin{matrix}(1.8062) \\{\left\{ \quad \right.\left( {1 - {WRF}} \right){V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}\left. \quad \right\}^{0.5}}\end{matrix}}{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.04}D^{0.5}}}\left( {T_{f} + T_{a}} \right)} \\{{{- \left( {1.3712 \times 10^{- 9}} \right)}\left( {T_{f} + T_{a}} \right)\left( {T_{f}^{2} + T_{a}^{2}} \right)\left( {T_{f} - T_{a}} \right)} + {(971)\overset{.}{w}}}\end{bmatrix}}\mspace{50mu} = {\frac{2}{\rho\; C_{p}s}{F\left( T_{f} \right)}}}} & \left( {21b} \right)\end{matrix}$Rewriting Eq. (21b) with time as the dependant variable and integratinggives,

$\begin{matrix}{{\int{\mathbb{d}t}} = {\left( \frac{\rho\; C_{p}s}{2} \right){\int\frac{\mathbb{d}T_{f}}{F\left( T_{f} \right)}}}} & \left( {21c} \right)\end{matrix}$Eq. (21c) could be used to get the facial temperature (T_(f)) at anytime (t) after exposure. Unfortunately, the integral on the right sideis not known to exist, at least at the time of this writing. Thus, thefacial temperature was determined through a numerical integration of Eq.(21b).

In Eq. (21b), the skin density (ρ) and the specific heat at constantpressure (C_(p)) were taken to be the same as those for water at afacial temperature of 91.4° F. (33° C.). Choosing the values to be thosefor water was based on a finding by Meyer (1971) that the productkρC_(p) for the skin should lie within the range of 15-60 Btu²/hr-ft⁴-°F.² (1.741×10⁶-6.964×10⁶ kg²/s⁵-° C.²), see Meyer, G. E., W., AnalyticalMethods in Conduction Heat Transfer, McGraw-Hill Book Company, pp. 202and 491, 1971. For water ρ=62.42 lbm/ft³ (999.84 kg/m³), C_(p)=1Btu/lbm-° F. (4.1868 KJ/kg-° C.) and the thermal conductivity k=0.36Btu/lbm-° F. (0.62 W/m-° C.). Consequently, the product kρC_(p)=22.47Btu²/hr-ft⁴-° F.² (2.608×10⁶ kg²/s⁵-° C.²) for water lies within theabove range. This value of the product, kρC_(p), is substantiated bysimilar values determined by Buettner, Yuan et al. and Valvano et al.,where their values were 14.98, 17.39 and 17.37 Btu²/hr-ft⁴-° F.²(1.739×10⁶, 2.019×10⁶ and 2.016×10⁶ kg²/s⁵-° C.², respectively). See,Yuan, D. Y., Valvano, J. W., Rudie, E. N. and Xu, L. X., “2-D finitedifference modeling of microwave heating in the prostate”,http://www/ece/utexas.edu/˜valvano/research/ASME95.pdf, 1995, andValvano, J. W., Nho, S. and Anderson, G. T., “Analysis of theWeinbaum-Jiji model of blood flow in the canine kidney cortex forself-heated thermistors”,http://www/ece/utexas.edu/˜valvano/research/ASME94.pdf, 1999. With thevalue of ρ=62.42 lbm/ft³ (999.84 kg/m³) and C_(p)=1 Btu/lbm-° F. (4.1868KJ/kg-° C.) substituted into Eq. (21b), this expression for thetemperature gradient when written in an incremental form becomes,

$\begin{matrix}{T_{f}^{m + {\Delta\; t}} = {T_{f}^{m} + {(1.83){\left( {\Delta\; t} \right)\left\lbrack \begin{matrix}{{K\left( {T_{CR} - T_{f}^{m}} \right)} + {({.65})G}} \\{{- \frac{(1.8062)\begin{Bmatrix}\left( {1 - {WRF}} \right) \\{V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}\end{Bmatrix}^{0.5}}{\begin{matrix}\begin{matrix}{{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.04}D^{0.5}} -} \\{\left( {1.3712 \times 10^{- 9}} \right)\left( {T_{f} + T_{a}} \right)}\end{matrix} \\{{\left( {T_{f}^{2} + T_{a}^{2}} \right)\left( {T_{f} - T_{a}} \right)} - {(971)\overset{.}{w}}}\end{matrix}}}\left( {T_{f} - T_{a}} \right)}\end{matrix} \right\rbrack}}}} & (22)\end{matrix}$With this equation, a step-by-step calculation of T_(f) can be made forany combination of T_(a) and V starting at time zero when m=0 and whereT_(j) ^(m)=91.4° F. (33° C.). Calculations with D=7 in. (0.5833 ft,0.1778 m) have shown that the optimum time increment Δt=1 sec.

(2) Infinite Series Approach: Eq. (22) represents what is referred to asa lumped capacitance approach to heat transfer. It is a preferred methodof treating transient conduction and indeed the only simple means whenmultiple modes of heat transfer exist. However, there is an errorassociated with a lumping of these multiple modes. It is small if theBiot number (Bi) here is defined as h_(fc)(s/2)/k≦0.1. This ratioimplies that if only an external forced convection heat loss waspresent, and none of the other heat transfer modes, it would initiate aone-dimensional conduction of heat through the skin thickness (s) drivenby the temperature difference (T_(CR)−T_(a)). With the known values forthe skin thickness (s) as 5.33 mm (0.0175 ft) and thermal conductivity(k) as 0.36 Btu/hr-ft-° F. (0.62 W/m-° C.) and with the forcedconvection heat transfer coefficient (h_(fc)) calculated from Eq. (13c),the Biot number can be calculated. With the Biot number, the magnitudeof the error associated with the lumped capacitance approach can bedetermined.

Since the h_(fc) is velocity (V) dependent, a calculation of Bi as afunction of velocity is shown in FIG. 3A when −140° F.≦T_(a)≦25° F.(−95.56° C.≦T_(a)≦−3.89° C.) and 0.1 mph≦V≦160 mph (0.16 Km/hr≦V≦257.49Km/hr). From FIG. 3A, the error associated with the lumped capacitanceapproach can be considered small if V≦5 mph (8.05 Km/hr). Since “small”is relative, and since the wind velocity will generally far exceed 5 mph(8.05 Km/hr), it was obvious that some other means would be necessary todetermine the error at the higher velocities. This was done bydetermining a time varying facial surface temperature (T_(f)) equationfor the above described one-dimensional analyses and comparing it withthe corresponding T_(f) equation from the lumped capacitance approach.The procedure uses an existing solution to the one-dimensional transientconduction problem.

The one-dimensional facial temperature expression uses the followingexisting exact infinite series solution disclosed in Incropera, F. P.and DeWitt, D. P., Introduction to Heat Transfer, John Wiley & Sons,Chap. 5, 1985,

$\begin{matrix}{\theta^{*} = {\sum\limits_{n = 1}^{n = \infty}{C_{n}{\exp\left( {{- \zeta_{n}^{2}}F_{o}} \right)}{\cos\left( {\zeta_{n}x^{*}} \right)}}}} & \left( {23a} \right)\end{matrix}$With T as the skin temperature at any axial location (x) in thedirection of heat flow, θ is the temperature difference (T−T_(a)) withT_(a) being the ambient temperature. θ* is the temperature difference θnormalized by the maximum temperature difference (T_(CR)−T_(a)) whereT_(CR) is the core temperature (98.2° F. (36.78° C.)). Thus,

$\begin{matrix}{\theta^{*} = \frac{T - T_{a}}{T_{CR} - T_{a}}} & \left( {23b} \right)\end{matrix}$In Eq. (23a), the Fourier number, F_(o)=4αt/s² where α is the skin'sthermal diffusivity (k/ρC_(p)) and is shown plotted in FIG. 3B as afunction of exposure time (t). Also in Eq. (23a), x* is the distance xfrom the midpoint of the segment normalized by half the skin thickness,that is x*=2x/s, x*=0 is the skin midpoint location and x*=1 is at theskin's outer surface. The coefficient C_(n) is defined as,

$\begin{matrix}{C_{n} = \frac{4{\sin\left( \zeta_{n} \right)}}{{2\zeta_{n}} + {\sin\left( {2\zeta_{n}} \right)}}} & \left( {23c} \right)\end{matrix}$where the discrete values (eigenvalues) of ζ_(n) are positive roots ofthe transcendental equation,ζ_(n) tan ζ_(n)=Bi  (23d)This infinite series solution can be approximated by the first term ofthe series solution when F_(o)≧0.2. FIG. 3B shows that this first termseries solution is applicable to all exposure times greater than 0.16min. A four term series solution was determined using the first fourroots of this equation as given in Appendix B.3 of Incropera et al. toobtain the normalized temperature.

The normalized mid-segment temperature is,

$\begin{matrix}{\theta_{o}^{*} = {\frac{\theta\left( {t,{x^{*} = 0}} \right)}{\theta_{i}} = \frac{{T\left( {t,{x = 0}} \right)} - T_{a}}{T_{CR} - T_{a}}}} & \left( {23e} \right)\end{matrix}$which yields the mid-segment temperature as,T _(x*=o) =T _(o)=θ_(o)*=(T _(CR) −T _(a))+T _(a)  (23f)The normalized surface temperature is,

$\begin{matrix}{\theta^{*} = {\frac{\theta\left( {t,{x^{*} = 1}} \right)}{\theta_{i}} = \frac{{T\left( {t,{x = {s/2}}} \right)} - T_{a}}{T_{CR} - T_{a}}}} & \left( {23g} \right)\end{matrix}$which yields the surface or facial temperature as,T _(x*=1) =T _(f)=θ*(T _(CR) −T _(a))+T _(a)  (23h)Equation (23h) is the facial temperature equation based on an infiniteseries heat conduction solution. Facial temperatures using this equationwere compared with those of Eq. (22) to determine if a significant erroris encountered as a result of using the lumped capacitance approach.

Time to Freeze Equation: In addition to providing a more preciseprediction of the wind chill temperature (T_(wc)), a key feature of thewind chill model of the present invention is its capability ofdetermining the exposure time when facial freezing will occur. The windchill temperature, although a true sensed temperature, becomessubjective in nature, since individuals may differ greatly in theirperception of its actual magnitude. When the time to freeze (t_(f)) isspecified, along with the wind chill temperature, it frees theindividual from becoming overly concerned with the actual value of thetemperature. Instead it becomes a warning to the individual of alimiting time after exposure when facial freezing will occur. It has theadded benefit of allowing preplanning of outdoor activities, so as notto exceed this time limit.

Ideally the dependence of facial temperature (T_(f)) on time (t) wouldbe obtained through integration of Eq. (21c) to get a closed formsolution for the time to freeze (t_(f)). Unfortunately, the integralshown in Eq. (21c) does not exist. Instead, the time to freeze (t_(f))was determined from Eq. (22) through step-by-step calculations of thetime for the facial temperature to decrease from the initial value of91.4° F. to 32° F. (33° C. to 0° C.). In this manner, considerable datawas generated on the time to freeze. This data was then curve fitted toobtain an explicit equation for the time to freeze (t_(f)). In thegeneration of this data, the evaporative heat loss (q_(e)) and themetabolic heat gain (q_(b)) were neglected in Eq. (22) since indicationswere that their effects would be offsetting. Only the primary heatlosses (q_(fc), q_(r)) were considered along with the solar heat gain(q_(i)), where the latter is based on values of G≧0 depending upon thepresence or absence of sunshine. As a further simplification, sea levelaltitude (H=0 ft (0 m)) was assumed with the intent of developing asubsequent correction to t_(f) for any other altitude.

Four equations for t_(f) were derived, one when sunshine is absent (G=0)and three others when sunshine is present (G>0), the latter three valuesof G being those for the latitude regions of Table 2, above.Calculations were made over the widest range of ambient temperature(−140° F.≦T_(a)≦25° F. (−95.56° C.≦T_(a)≦−3.89° C.)) and wind velocities(0 mph≦V≦160 mph (0 Km/hr≦V≦257.49 Km/hr)) anticipated worldwide. Thefollowing procedure was used:

-   -   (1) With G=0 and T_(a)=−140° F. (−95.56° C.), calculate the        facial temperature (T_(f)) decay curves for values of V=0.1, 1,        5, 10, 20, 40, 60, 80, 100, 120, 140 and 160 mph (0.16, 8.05,        16.09, 32.19, 64.37, 95.56, 128.74, 160.93, 193.12, 225.30 and        257.49 Km/hr).    -   (2) Plot the T_(f) decay curves as shown in FIGS. 4A-B and note        the times to freeze (t_(f)) as their intercept points with the        T_(f)=32° F. (0° C.) line. FIG. 4B is an enlarged graph of the        region in dotted ellipse shown in FIG. 4A.    -   (3) Along the line T_(f)=32° F. (0° C.), obtain values of t_(f)        vs. V and plot t_(f) vs. log V as a dashed line in FIG. 5 for        T_(a)=−140° F. (−95.56° C.).    -   (4) Repeat steps (1)-(3) for each of the other values of T_(a),        −120° F., −100° F., −80° F., −60° F., −40° F., −20° F., 0° F.,        10° F., 20° F. and 25° F. (−84.44° C., −73.33° C., −62.22° C.,        −51.11° C., −40° C., −28.89° C., −17.78° C., −12.22° C.,        −6.67° C. and −3.89° C.) to get the remaining dashed curves in        FIG. 5.

The eleven dashed curves of FIG. 5 were curve fitted using TableCurve3D™ to obtain the following equation for the freezing time (t_(f)) whenG=0:

$\begin{matrix}{t_{f} = \frac{\left\{ {a + {b\mspace{14mu}{\ln(V)}} + {c\left\lbrack {\ln(V)} \right\rbrack}^{2} + {d\left\lbrack {\ln(V)} \right\rbrack}^{3} + {e\; T_{a}}} \right\}}{\left\{ {1 + {f\mspace{14mu}{\ln(V)}} + {g\left\lbrack {\ln(V)} \right\rbrack}^{2} + {h\; T_{a}} + {i\left( T_{a} \right)}^{2}} \right\}}} & (24)\end{matrix}$where a=12.213472, b=−4.7287903, c=0.71377035, d=−0.040374904,e=0.0020284147, f=−0.042770886, g=0.0048025828, h=−0.016212428,i=−2.3703809×10⁻⁵ and where the correlation coefficient isr²=0.99984977. The curve fitted values of t_(f) vs. log V using Eq. (24)are shown as solid lines in FIG. 5 and show a good correlation with thecalculated values shown as dashed lines. These curves show that in theabsence of sunshine (G=0), the time to freeze decreases very rapidly asthe velocity increases, especially at the higher temperatures. Becausethe presence of sunshine (G>0) has been shown to even prevent freezing,these curves would be expected to shift progressively upward as thevalue of G increases. Because of this pronounced effect of sunshine ont_(f), the above calculations and curve fits were made for each of thethree regional values of G in Table 2. The different set of constants(a, b, c, d, e, f, g, h, i) developed for each value of G are shown inTable 3, below:

TABLE 3 Coefficients in Eq. (24) for the time to freeze (t_(f)) in threelatitude regions Latitude Region 45°-50° 40°-45° 35°-40° All Latitudes G= 37.78 Btu/hr- G = 42.66 Btu/hr-ft² G = 58.51 Btu/hr- Constants G = 0ft² (119.17 W/m²) (134.58 W/m²) ft² (184.58 W/m²) a 12.213472 18.18596618.510193 27.201736 b −4.7287903 −7.1895224 −7.96409 −10.167441 c0.71377035 1.1058166 1.3362907 1.418616 d −0.040374904 −0.065135256−0.084835215 −0.07397277 e 0.0020284147 −0.0001595948 −0.0026299421−0.0014226544 f −0.042770886 0.092286021 0.011675974 0.36636446 g0.0048025828 −0.013081903 −0.0021834205 −0.048107247 h −0.016212428−0.02597853 −0.024713305 −0.042429888 i −2.3703809 × 10⁻⁵ −3.1495047 ×10⁻⁵ −1.5165097 × 10⁻⁵ 5.740629 × 10⁻⁵ r² 0.99984977 0.999014590.99670162 0.99766982

As already noted, it was essential to use Eq. (22) in the determinationof Eq. (24) because the integral in Eq. (21c) was nonexistent. If theintegral had consisted of only one rather than five heat loss/gainterms, a closed form solution of Eq. (21c) would have been possible.This means that if either of the primary heat losses (radiation, forcedconvection) is present, a closed-form solution for the time to freeze(t_(f)) is possible.

Closed Form Solution of t_(f)-Radiation Only: When an individual isexposed to a cold environment, a radiation heat loss will always existwhen T_(f)>T_(a) even in the absence of a forced convection heat loss(V=0 mph (0 Km/hr)). Considering only the radiation heat loss as definedin Eq. (2), Eq. (21c) becomes,

$\begin{matrix}{{\int_{t_{i}}^{t_{fr}}{\mathbb{d}t}} = {\left( \frac{\rho\; C_{p}s}{2} \right){\int_{T_{fi}}^{T_{ff}}\frac{\mathbb{d}T_{f}}{{ɛ\sigma}\left( {T_{a}^{4} - T_{f}^{4}} \right)}}}} & \left( {25a} \right)\end{matrix}$Integrating from the initial exposure time, t_(i)=0, when the initialfacial temperature is T_(fi)=91.4° F. (551.09° R) to the time to freeze,t_(fr), when the final facial temperature T_(ff)=32° F. (491.69° R)gives,

$\begin{matrix}{t_{fr} = {\left( \frac{\rho\; C_{p}s}{2{\sigma ɛ}} \right)\left\lbrack {{\frac{1}{4T_{a}^{3}}\ln{\frac{T_{a} + T_{f}}{T_{a} - T_{f}}}} + {\frac{1}{2T_{a}^{3}}{\tan^{- 1}\left( \frac{T_{f}}{T_{a}} \right)}}} \right\rbrack}_{T_{fi}}^{T_{ff}}} & \left( {25b} \right)\end{matrix}$Inserting the numerical values for the facial temperature limits intoEq. (25b) gives,

$\begin{matrix}{t_{fr} = {\left( \frac{\rho\; C_{p}s}{2{\sigma ɛ}} \right)\left( \frac{1}{2T_{a}^{3}} \right)\begin{Bmatrix}{{\frac{1}{2}\left\lbrack {{\ln{\frac{T_{a} + 491.69}{T_{a} - 491.69}}} - {\ln{\frac{T_{a} + 551.09}{T_{a} - 551.09}}}} \right\rbrack} +} \\{{\tan^{- 1}\left( \frac{491.69}{T_{a}} \right)} - {\tan^{- 1}\left( \frac{551.09}{T_{a}} \right)}}\end{Bmatrix}}} & \left( {25c} \right)\end{matrix}$With the values of ρ, C_(p), s, ε, and σ already known, Eq. (25c)provides an exact value for the times to freeze, t_(f), at any ambienttemperature (T_(a)) when only the radiation heat loss is present.

Closed Form Solution of t_(f)-Forced Convection Only: A forcedconvection heat loss is not likely to exist in the absence of aradiation heat loss. However, if it does, and with the convection heatloss as defined by Eq. (1), Eq. (21c) becomes,

$\begin{matrix}{{\int_{t_{i}}^{t_{fc}}{\mathbb{d}t}} = {\left( \frac{\rho\; C_{p}s}{2} \right){\int_{T_{fi}}^{T_{ff}}\frac{\mathbb{d}T_{f}}{h_{fc}\left( {T_{f} - T_{a}} \right)}}}} & \left( {26a} \right)\end{matrix}$Integrating as before gives,

$\begin{matrix}{t_{fc} = {\left( \frac{\rho\; C_{p}s}{2\; h_{fc}} \right){\ln\left\lbrack {T_{f} - T_{a}} \right\rbrack}_{T_{fi}}^{T_{ff}}}} & \left( {26b} \right)\end{matrix}$Inserting the numerical values for the facial temperature limits intoEq. (26b) along with the expression for h_(fc) from Eq. (13c) gives,

$\begin{matrix}{t_{fc} = {\left( \frac{\rho\; C_{p}s}{2\;} \right)\;\frac{\;{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.04}D^{0.5}}}{(1.8062)\left\{ {\left( {1 - {WRF}} \right){V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}} \right\}^{0.5}}{\ln\left( \frac{551.09 - T_{a}}{491.69 - T_{a}} \right)}}} & \left( {26c} \right)\end{matrix}$With values of ρ, C_(p), s and D known and ambient conditions (T_(a), V)specified along with altitude (H) and wind reduction factor (WRF), Eq.(26c) provides an exact value for the time to freeze when only aconvection heat loss is present.

Wind Chill Temperature: Comparisons were made between the predictions ofthe wind chill temperatures using Eq. (18a) and the Siple and Passelresults. The Siple and Passel experiment was conducted at or near sealevel (H=0) conditions at Little America, Antarctica. Because theirexperiment was conducted out in the open, well above ground level, therewas no reduction in the wind speed (WRF=0). Neither the solar heat gain(G=0) nor the evaporative heat loss ({dot over (w)}=0) were of concern,since Siple and Passel, when referring to insolation and evaporation,stated that these factors “were almost missing in the Antarctic winterwhen the experiments were carried on”. Finally, the metabolic heat gainwas not present (q_(b)=0) because it did not exist in their experiment.Since the radiation heat loss term in Eq. (18a) and the experiment areidentical, except for an emissivity factor, any major difference in thewind chill temperatures as predicted by Eq. (18a) and the Siple andPassel results would be due to a difference in the forced convectioncoefficients. Determining this difference meant extracting the Siple andPassel forced convection coefficient from their test results andcomparing it with the forced convection coefficient of Eq. (13c) as usedin this model. This is described in the following paragraphs.

Determination of the inherent forced convection coefficient from theSiple and Passel results began with an equation that expressed theirresults as wind chill temperatures. This equation,T _(wc)=91.4−(0.04544)[10.45−0.447V+6.6858V ^(0.5)](91.4−T _(a))  (27a)was obtained from Marks' Standard Handbook for Mechanical Engineers, 9thed., McGraw-Hill, Chap. 12, p. 113, 1989, where it is shown as Eq.(12.4.47). In this equation, V is in mph and T_(wc) and T_(a) are in °F. This equation can be written in terms of what has been called thewind chill index (WCI), which represents the heat loss rate ({dot over(q)}) from their test container when the container surface temperaturewas assumed to be at a constant 91.4° F. (33° C.). Although Siple andPassel erred in their assumption that this temperature of 91.4° F.remained constant, a simple test using a hand held thermometer incontact with the skin will verify that this was a good choice for theinitial skin temperature of a human suddenly exposed to a coldenvironment. Their so-called wind chill index (WCI) from Eq. (27a) is,WCI_(E)=[10.45−0.447V+6.6858V ^(0.5)](91.4−T _(a))={dot over (q)}_(E)  (27b)and has English units of Btu/hr-ft². It is shown as Eq. (12.4.45) inMark's. Siple and Passel, who initiated the term WCI, expressed it as,WCI_(SI)=[10.45−V+10V ^(0.5)](33−T _(a))={dot over (m)} _(SI)  (27c)where it has SI units in Kcal/hr-m² and is shown as Eq. (12.4.44) inMark's. Equation (27a), as shown in Eq. (12.4.47), can be expressed interms of WCI_(E) as,T _(wc)=91.4−0.04544(WCI_(E))  (27d)and its counterpart in SI units from Eq. (12.4.44) in Mark's can bewritten as,T _(wc)=33−0.04544(WCI_(SI))  (27e)Since there is an equivalency in T_(wc) irrespective of the units inwhich it is expressed, then from Eqs. (27d) and (27e), WCI_(E)=WCI_(SI).This means that {dot over (q)}_(E)={dot over (q)}_(SI) where from Eqs.(27b) and (27c),{dot over (q)} _(E)=[10.45−0.447V+6.6858V ^(0.5)](91.4−T _(a))=h_(E)(91.4−T _(a))  (27f)and,{dot over (q)} _(SI)=[10.45−V+10V ^(0.5)](33−T _(a))=h _(SI)(33−T_(a))  (27g)

These two equations demonstrated that over the same temperature range,h_(E) must be made equivalent to h_(SI) through a unit conversionfactor. To convert from SI units in which the Siple and Passel resultsare expressed in English units, h_(E) must be multiplied by 1/4.88 or0.2049. Thus, the actual Siple and Passel WCI or heat flow rate is notthe expression in Eq. (27b) but rather,WCI=(0.2049)[10.45−0.447V+6.6858V ^(0.5)](91.4−T _(a))  (27h)The presence of this coefficient (0.2049) indicates that Eq. (12.4.45)in Mark's is in error. It also changes the existing coefficient (0.5556)in the WCI expression in Eq. (69) of 1993 ASHRAE Handbook, Fundamentalsto 0.1138. Consider the case where V=4 mph (1.79 m/s) and T_(a)=0° F.(−17.78° C.). The error in Mark's will be demonstrated based on the heatflux rate followed by the convective heat transfer coefficientequivalencies. Substituting the values in English units in Eq. (27b)yields {dot over (q)}_(E)=2013.87 Btu/hr-ft² (6352.95 W/m²). Similarly,substituting the values in SI units in Eq. (27c) yields {dot over(q)}_(SI)=1119.15 Kcal/hr-m² or 1301.32 W/m² (412.64 Btu/hr-ft²). Theyare equivalent when Eq. (27b) is multiplied by a factor of 0.2049.Similarly, substituting the values in English units in Eq. (27f) yieldsh_(E)=22.03 Btu/hr-ft²-° F. (125.10 W/m²-° C.). Substituting the valuesin SI units in Eq. (27g) yields h_(SI)=25.63 W/m²-° C. (4.51Btu/hr-ft²-° F.). They are equivalent when Eq. (27f) is multiplied by afactor of 0.2049.

Eq. (27h) represents the actual heat flow from the Siple and Passelcontainer. Eq. (27h) may be expressed as,WCI=h _(fc,r)(91.4−T _(a))  (27i)where h_(fc,r) is a combined forced convection and radiation heattransfer coefficient measured during their tests and where from Eq.(27h),h _(fc,r)=(0.2049)[10.45−0.447V+6.6858V ^(0.5)]  (27j)Since the radiation coefficient in the Siple and Passel tests wasexpected to take on the same form as that in Eq. (14c) for the humanhead, their radiation coefficient, assuming their emissivity of ε=1rather than ε=0.8, was,h _(r)=(1.714×10⁻⁹)(91.4+T _(a))└(91.4)² +T _(a) ²┘  (27k)Consequently, the Siple and Passel forced convection coefficient(h_(fc)) was determined from,h _(fc) =h _(fc,r) −h _(r)  (271)where upon using Eqs. (27j) and (27k), it became,h _(fc)=(0.2049)└10.45−0.447V+6.6858V ^(0.5)┘−(1.714×10⁻⁹)(91.4+T_(a))[(91.4)² +T _(a) ²]  (27m)

A comparison was made of the forced convection coefficient values usingthis deduced expression of h_(fc) from the Siple and Passel experimentwith the corresponding values from the analytical expression (h_(fc))shown in Eq. (13c). The comparison was made for an ambient temperature(T_(a)) of 0° F. (459.69° R) over the velocity range 0<V<45 mph(0<V<72.42 Km/hr) that existed during the Siple and Passel experiment.In addition to the previously mentioned conditions (WRF=H=0) inAntarctica, the surface temperature (T_(s)) in Eq. (13c) was set at91.4° F. (33° C.) and the diameter D at 2.26 in (0.1883 ft, 0.0574 m) soas to match the container diameter used in the Siple and Passelexperiment. Calculated values of the two h_(fc) coefficients are plottedin FIG. 6A and the results are surprising. The Siple and Passel values(curve 1) are but a small percentage of this model's values (curve 2)over the entire velocity range. Included in FIG. 6A are two curves fromFIG. 1 of Osczevski's paper. Curve 3 represents Osczevski's test resultsfor a thermal model of the head which is assumed to have a diameterD=0.16 m (6.3 in). Curve 4 represents the results of Osczevski'scalculation of the coefficient for the test cylinder (D=2.26 in. (0.1883ft, 0.0574 m)) used by Siple and Passel. In making this calculation,Osczevski used Eq. (6) in his paper which he calls a standard empiricalequation for h_(fc) for a cylinder. The good agreement betweenOsczevski's predicted values of h_(fc) for the Siple and Passelcontainer (curve 4) and his own test results (curve 3) may have ledOsczevski to state that the cylinder used by Siple and Passel was nearlya perfect representation of the head. This statement indicates thatOsczevski may have believed that his calculated value of h_(fc) for theSiple and Passel container actually existed in their experiment. Butthis was not the case, because the actual Siple and Passel values ofh_(fc) (curve 1) are much less than Osczevski's predicted values oftheir h_(fc) (curve 4). In fact Osczevski's predicted values of h_(fc)are even much greater than the total or combined convection/radiationcoefficients (h_(fc,r)) in the Siple and Passel tests as represented bycurve 5. Based on these findings, it appears that the measured containercooling rates in the Siple and Passel experiment were much too low. Thereason for this is unclear. However, Siple and Passel in FIG. 4 of theirpaper note several problems encountered during their experiment, such asanemometer difficulties, turbulence, and local convection currents, allpossibly leading to unreliable data.

From FIG. 6A, it is evident that the prediction of h_(fc) (curve 2)using Eq. (13c) results in an increasing difference between it and theh_(fc) from the Siple and Passel test data (curve 1) as the velocityincreases. This difference reaches a maximum at V=45 mph (72.42 Km/hr)where the h_(fc) from Eq. (13c) is 3.4 times greater than that for thetest data. At this same velocity, the h_(fc) calculated from Osczevski'sEq. (6) is 2.4 times greater. Although Eq. (13c) and Osczevski's Eq. (6)were derived from the same basic equation, h_(fc)=Nu_(D)k/D=CRe_(D))^(m)(Pr)^(n)(k/D), Osczevski's choice for the coefficient (0.23)and the exponent (0.6) in Eq. (6) limits its applicability to a Reynoldsnumber range of 4,000 to 40,000. But in reality, the Reynolds number mayfar exceed this range. For example, with the head diameter D=7 in.(0.5833 ft, 0.1778 m) and a velocity V=45 mph (72.42 Km/hr), theReynolds number is 257,000, which is well beyond the useful range of Eq.(6). Equation (13c) is free of this restriction, being adaptable overthe entire range of anticipated Reynolds numbers, and most likelyrepresents what the real coefficients were in the Siple and Passeltests.

FIG. 6A shows that this model's predicted forced convection coefficients(curve 2) for the Siple and Passel container were as much as 3.4 timeslarger at V=45 mph than those (curve 1) derived from their test results.Most likely due in part to problems they had in the recording of theirdata, this difference in the values of the forced convectioncoefficients means that the Siple and Passel values of the wind chilltemperature would be higher (warmer) over the entire velocity range, butespecially at the higher velocities. This confirms the previousstatement that any difference in these predicted values of the windchill temperature would be due to differences in the forced convectioncoefficients. This is demonstrated in FIG. 6B, where the Siple andPassel values of T_(wc) from Eq. (27a), superimposed with the usuallytabulated Siple and Passel results (▴), are much higher (warmer) thanthose calculated from Eq. (18a) using the previously mentioned Antarcticconditions. For example, at T_(a)=0° F. (−17.78° C.) and V=45 mph, Eq.(18a) predicts a wind chill temperature that is −123° F. (−68.33° C.)colder than the Siple and Passel value. What should be noted is thatthis difference is much less at the lower velocities. These comparisonscan only be made over the ambient temperature range (−40° F.≦T_(a)≦40°F. (−40° C.≦T_(a)≦4.44° C.)) associated with the Siple and Passelexperiment, whereas the calculations with this model can span a muchbroader range of ambient temperature (−140° F.≦T_(a)≦91.4° F. (−95.56°C.≦T_(a)≦33° C.)), over which wind chill temperatures can be of realconcern.

FIG. 7A shows a comparison of the Siple and Passel coefficients (curve1) with the wind chill model's prediction (curve 2) of the presentinvention for the head with D=7 in. Curve 2 shows a large decrease inthe predicted value of h_(fc) as the diameter is increased from 2.26 in.(5.74 cm) in FIG. 6A to 7 in. in FIG. 7A. As a result, the predictedvalue of h_(fc) at T_(a)=0° F. and V=45 mph is now only 1.9 times, not3.4 times, larger than the Siple and Passel value. This means that thelarger difference shown in FIG. 6A was not due entirely to unreliabilityin the Siple and Passel data, but is also due to the fact that theforced convection coefficient varies inversely with the diameter. Thissuggests that if Siple and Passel had chosen a container with a largerdiameter, closer to that of the human head, their test results may haveshown values of h_(fc) (curve 1) much closer to those predicted by thewind chill model of the present invention (curve 2). Nevertheless, thecurves of FIG. 7A reflect the real differences that currently exist.

The wind chill temperature was then calculated for the head (D=7 in.)using the h_(fc) (curve 2) from FIG. 7B. FIG. 7B shows the differencebetween the wind chill temperature as predicted by the wind chill modelof the present invention for the head with those obtained by Siple andPassel. The model predicts at T_(a)=0° F. and V=45 mph a wind chilltemperature that is −40° F. (−22° C.) colder than that of Siple andPassel. At the same temperature, the wind chill model of the presentinvention predicts a temperature that is only −5.2° F. (−2.9° C.) colderat V=10 mph (16.09 Km/hr). The fact that the Siple and Passel values ofT_(wc) are only moderately warmer than the predicted values at the lowervelocities may partly explain the usefulness of their values over thedecades since they conducted their experiment. The fact that the Sipleand Passel values of T_(wc) are warmer than the actual values iscontrary to previous criticism suggesting the opposite. This criticismwas due to Siple and Passel's assumption that the skin temperature of91.4° F. (33° C.) would remain constant during the time of exposure.This was a valid criticism, since this skin temperature would decreasefrom its initial value of 91.4° F. with increasing exposure timeresulting in a decreased T_(wc) as well. However, this criticism doesnot preclude the usefulness of the Siple and Passel values because theyrepresent the maximum wind chill temperature an individual wouldexperience during a very short exposure to the wind. This maximum valuemight be all the wind chill information an individual needs to refrainfrom a more sustained exposure that might lead to facial freezing.

Facial Temperature: No direct comparison of facial temperature aspredicted by Eq. (22) was possible since no other model predictions ofthese temperatures were available. However, the predicted values showedgood agreement with several sets of test data. The initial interest inthe facial temperature stemmed from Siple and Passel's assumption of aconstant body surface temperature which limited the determination ofwind chill to individuals momentarily exposed to the elements. It doesnot apply when individuals are subjected to an extended exposure whenthe surface temperature must necessarily decrease. Bluestein and Zecherin their calculations of a more accurate wind chill considered the timedependency of this temperature. Aside from their correction of thisconstant surface temperature, Bluestein and Zecher presumably made othercorrections to what they and numerous investigators have referred to asthe flawed Siple and Passel experiment. The inference was that theseflaws introduced error or somehow invalidated the experiment's results.Looking at each of these flaws, the simple human error of misidentifyingthe heat transfer coefficient as a cooling rate does nothing toinvalidate the results. Neither does a particular curve fit of the datainvalidate the data upon which it was based. Granted the data itself hadinaccuracies, which seem to have been due to conditions beyond thecontrol of the experimenters. Furthermore, it was stated that Siple andPassel ignored thermal gradients through the water and the container anddid not consider the difference between radiation and convection heatloss. It should be obvious that in their experiment these gradientsnaturally developed, and that the radiative and convective heat transferactually took place as part of the whole thermal process of containercool-down. The fact that these gradients and heat losses were aninherent part of the experiment makes the claim that they were notconsidered totally irrelevant. One concludes from all this that the onlyvalid flaw was Siple and Passel's assumption of a constant bodytemperature during the entire exposure time.

Bluestein and Zecher's development of a more accurate wind chill modelwas intended to correct the flaws in the Siple and Passel experiment.They recognized the fact that Siple and Passel's assumption of aconstant 91.4° F. (33° C.) facial temperature would lead to a lower(colder) wind chill temperature than that which would actually exist.Their iterative heat transfer analysis of the surface of a cylindersimulating the human head provided a means of computing this temperatureas it decreased with exposure time. Consequently, their results wouldhave been expected to yield considerably higher (warmer) wind chilltemperatures than those of Siple and Passel. But this was not the case.This can be verified by a comparison of the tabulated results inBluestein and Zecher's paper. Table 1 of their paper shows the Siple andPassel wind chill temperature from Eq. (27a) based on their assumptionof a constant 91.4° F. facial temperature. Table 2 of their paper showsthe final Bluestein and Zecher results based on a varying facialtemperature. These Table 2 results also include a wind reduction factor(WRF) of 0.33 to account for the fact that the NWS 10 m velocity wasassumed to be 50% greater than that at head level. The portion of theseTable 2 results that can be attributed to the varying facial temperaturecan be determined by removing the wind reduction effects from theseresults. This can be done for a given temperature (T_(a)) and velocity(V) by using the WRF of 0.33 to reduce the velocity in Eq. (27a) andthen comparing the computed wind chill temperature with thecorresponding value at the same T_(a) and V in Table 2. The differencein these values reflects what Bluestein and Zecher found as an increaseor moderation in the wind chill temperature as a result of the facialtemperature variation. For example, consider the case where T_(a)=−40°F. and V=40 mph. Using a WRF=0.33 in Eq. (27a) yields a Siple and Passelvalue of T_(wc)=−105.97° F. The corresponding Bluestein and Zecher valuein their Table 2 is T_(wc)=−106.7° F. This slight difference of −0.73°F. is not even a moderation but rather a decrease in the wind chilltemperature. For all other combinations of T_(a) and V, the Bluesteinand Zecher results show, at most, a 2° F. moderation in the wind chilltemperature. Consequently, without any wind reduction, the Siple andPassel and Bluestein and Zecher results are nearly identical, asillustrated in FIG. 8.

This means that the Bluestein and Zecher results reflect a negligiblemoderation in the wind chill temperature due to a decreasing facialtemperature. Assuming Bluestein and Zecher's calculations of thedecreasing facial temperature are correct, then their choice of theChurchill-Bernstein forced convection equation may have been the reasonthat their results show this negligible moderation. Bluestein (1998)chose the Churchill-Bernstein equation because it was appropriate forthe Reynolds and Prandtl numbers that existed during the Siple andPassel experiment, see Bluestein, M., “An evaluation of the wind chillfactor: its development and applicability”, J. of Biomech. Eng., Vol.120, pp. 255-258, 1998. Perhaps the choice of the Churchill-Bernsteinequation was influenced by the fact that this equation yields values ofthe forced convection coefficient that are nearly identical to those ofEq. (6) in Oscevski's paper. Oscevski had already shown that in applyingEq. (6) to the Siple and Passel cylinder he achieved good agreement withhis own test data as shown by curves 3 and 4 in FIG. 6A. But curve 4shows Oscevski's prediction of what the Siple and Passel coefficientsshould have been and not what they actually were (curve 1).Consequently, had Bluestein and Zecher used the Siple and Passelcoefficient (curve 1) rather than the Churchill-Bernstein equation intheir analyses, they might well have seen the real effect of decreasingfacial temperature on T_(wc) in their Table 2 results. Choosing theChurchill-Bernstein equation actually resulted in convectioncoefficients that were significantly higher than those in the Siple andPassel experiment. This resulted in an even greater increase inconvective cooling, relative to the actual Siple and Passel cooling(curve 1) and may have resulted in a wind chill temperature decreasethat was almost equal to the wind chill temperature increase due to thefacial temperature decrease. If this was the case, then Bluestein andZecher's usage of the Churchill-Bernstein equation and possibly correctcalculations of the time varying facial temperature ended up withessentially the same wind chill temperatures as Siple and Passel did 60years ago. The only moderating effect shown by Bluestein and Zecher'sresults is that due to their incorrect assumption of a constant WRF forall combinations of ambient temperature (T_(a)) and wind velocity (V).This blanket assumption is tantamount to saying that the same WRF of0.33 applies to each and every individual throughout the entire weatherreport listening area. As shown below, this is a crucial error since aWRF=0.33 will almost never occur. The fact that the Bluestein and Zecherresults which show virtually no change in the wind chill temperaturewith exposure time, meant that their facial temperature results wouldnot provide the expected comparison with the definite facial temperaturedecrease predicted with Eq. (22).

Predicted facial temperatures using Eq. (22) were verified using thecomputed and measured facial temperatures found by Adamenko andKhairullin, by Buettner (1951), by LeBlanc, J., Blais, B., Barabe, B.and Cote, J., “Effects of temperature and wind on facial temperature,heart rate, and sensation”, J. Appl. Physiology, Vol. 40, No. 2, pp.127-131, 1976, (hereinafter, LeBlanc et al. (1976)), by the inventorsthrough home freezer experiments and those found during a DiscoveryChannel™ experiment. The previously discussed experiment by Adamenko andKhairullin, which was conducted in northern Russia, was extensive inthat it consisted of instrumenting three facial components (cheeks,nose, ears) on 40 people during their exposure to temperatures (T_(a))ranging from 10° C. to −40° C. (50° F. to −40° F.) and wind velocities(V) up to 15 m/s (33.55 mph). The temperature of each component after ahalf to one hour was expressed in terms of the ambient temperature andwind velocity. In each case, the component temperature was 33° C. (91.4°F.) prior to exposure. These test data provided information on facialtemperature variation and the ambient conditions when facial freezingwould occur. This information was expected to be useful in determiningthe exposure time when facial freezing occurs, since this facialfreezing time is a key element in our model that makes the concept ofwind chill temperature less subjective.

The results of the Adamenko and Khairullin experiment were described bySchaefer, J. T., “The effect of wind and temperature on humans”,National Weather Service, Central Region Technical Attachment 88-05, pp.1-2, 1988, as further described in Schwerdt, R. W., “Letters to theeditor”, Bull. Amer. Meteor. Soc., Vol. 76, No. 9, pp. 1631-1636, 1995.Schwerdt made a convincing argument for informing the listening publicof the relationship between the wind chill temperature and facialfreezing. The Adamenko and Khairullin results were presented as a leastsquare equation of all measured temperatures for each of the threefacial components in terms of the ambient conditions. An average ofthese equations provided an equation for the approximate facialtemperature (T_(f)) after a half to a one hour exposure to the ambientconditions. This equation is,T _(f)=0.4T _(a)−3.3V ^(0.5)+16  (28)where the temperatures (T_(f), T_(a)) are in ° C. and the wind velocity(V) is in m/s. As an example, this equation shows that if the face is tocool down from 33° C. to a condition of facial frostbite (T_(f)=0° C.)within a one half to one hour exposure in calm conditions (V=0 m/s),then the ambient temperature must be T_(a)=−39.9° C. It should be notedthat Eq. (28) for the face temperature is similar to each of the threecomponent equations in that they all differ only in the value of theconstant. These values are 19, 17, and 12 for the cheek, nose, and ear,respectively; the lower the value, the more readily the componentfreezes. Adamenko and Khairullin concentrated their efforts ondetermining the nose temperature, because the nose is more difficult toprotect than the cheeks and the ears, and consequently is morevulnerable to freezing. The use of Eq. (28) for the face temperature,where the constant (16) is slightly less than that for the nose (17),provides a degree of conservatism in that it forewarns of thepossibility of freezing before it is likely to occur.

FIG. 9 is a plot of Eq. (28) for two ambient temperatures (−10° C. and−8.25° C. (14° F. and 17.15° F.)) and shows the relationship between thefacial temperatures as predicted by Adamenko and Khairullin with theexperimental results of Osczevski, LeBlanc et al. (1976), and Wilson,O., “Cooling effect of an Antarctic climate on man: With someobservations on the occurrence of frostbite”, Norsk Polarinstitutt,Skrifter N R. 128, pp. 26-30, 1963, (hereinafter, Wilson (1963)). Thesetemperatures (−10° C., −8.25° C.) were selected because they span thetemperature range of all three sets of experimental results. Considerthe case where the ambient temperature is −10° C. in Eq. (28). It showsthat the face at an initial temperature of 33° C. (91.4° F.), whenexposed to a wind speed of 13 m/s (42.65 ft/s), will cool down to thefacial freeze condition of 0° C. (32° F.) in a one half to one hourperiod. Under calm conditions (V=0 m/s), the face will cool down to 12°C. (53.6° F.) over the same time period. At this same ambienttemperature, (−10° C.), Osczevski's facial cooling model shows facialcooling to the 0° C. temperature at a reduced speed of 7.25 m/s (23.79ft/s). At the same ambient temperature (−10° C.) and at wind tunnelvelocities of approximately 6.0 m/s and 10 m/s (19.68 ft/s to 32.81ft/s), LeBlanc et al. (1976) obtained nose temperatures of 10° C. and7.5° C. (50° F. and 45.5° F.), respectively. These slightly highertemperatures may have been due to the relatively short 10 min. ofexposure time experienced by each of their subjects to the simulatedambient conditions at the wind tunnel exit. The three Wilson (1963) datapoints at 10, 15, and 20 m/s (32.81, 49.21 and 65.62 ft/s) are just afew of the 69 events reported by Wilson during aNorwegian-British-Swedish Expedition to Antarctica. Expressing theseresults in terms of the wind chill index (WCI) as defined by Siple andPassel in Eq. (27c), Wilson states that all these cases of freezing“have occurred at wind chill values between 1400 and 2100 Kcal/m²-hr(516.18 and 774.27 Btu/hr-ft²), at temperatures ranging from −8° C. to−46° C. (17.6° F. to −50.8° F.) which confirms the assumption of Sipleand Passel that exposed flesh begins to freeze at a wind chill index of1400 Kcal/m²-hr”. Their assumption is clearly confirmed by Wilson'sthree points at 10, 15 and 20 m/s where the corresponding facialtemperatures (T_(f)) are 1.56° C., −0.4° C. and −2.06° C. (34.81° F.,31.28° F. and 28.29° F.) and where all three of these temperatures areclose to the freezing condition (0° C. (32° F.)) and all the points lieclose to the superimposed WCI=1400 Kcal/m²-hr line. Osczevski concurs bynoting that at this value of the index there is “danger of facialfrostbite”. One concludes from this that the facial temperatures basedon the Adamenko and Khairullin test results are in good agreement withthose obtained by Osczevski, Wilson (1963) and LeBlanc et al. (1976). Asalready noted by Schwerdt and as evidenced by one of the Wilson datapoints, actual facial freezing occurs closer to −2° C. (28.4° F.) thanthe presumed value of 0° C. This was remarkably demonstrated in December2002 when American ocean swimmer Lynne Cox, clad only in a swimsuit,cap, and goggles, successfully swam 1.22 miles (1.96 Km) in 0° C. wateroff the coast of Antarctica without suffering any freezing of the skin.By assuming the 0° C. freezing point in our model, an additional degreeof conservatism is added in that a warning of facial freezing can bemade prior to its actual occurrence.

Equation (28) is expressed as follows in English units so as to put itin a more practical form for facial temperature calculations,T _(f)=0.4(T _(a)−32)−3.97V ^(0.5)+60.8  (29)where in this equation, T_(f) and T_(a) are in ° F. and V is in mph. Theinitial facial temperature upon exposure to the wind is now 91.4° F.(33° C.). Equation (29) represents the face temperature after a half toone hour cool down from 91.4° F. at ambient conditions (T_(a), V).Schwerdt notes that “The higher the wind speed, the faster (closer to ahalf hour than an hour) the skin will freeze, assuming all conditionsexcept wind speed are equal”. Consequently, for the higher wind speedsof interest in wind chill predictions, the facial temperature T_(f) inEq. (29) was assumed to be reached after a half hour of exposure. Itshould also be emphasized that although Eq. (29) permits the calculationof the face temperature after 30 min., it cannot be used to determinethe temperature at some other time within this 30 minute interval. Thisis because the actual variation of the facial temperature with time isnot linear as one might infer in an application of Eq. (29). Rather itis nonlinear and because of this nonlinearity, Eq. (22) for the surfacetemperature was developed using the conservation of energy principle.Before comparing the computed values of the facial surface temperature(T_(f)) from Eq. (22) with the Adamenko and Khairullin test results, acomparison was made with four other studies and experiments.

Buettner Study: A study by Buettner (1951) determined the temperaturedecay of warm bare skin when impacted by a cold stream of air. Hisexpression for the decrease in temperature, here expressed as a facialtemperature, is the classic solution for surface convection over asemi-infinite solid, that is,

$\begin{matrix}{T_{f} = {T_{CR} - {\left( {T_{a} - T_{CR}} \right)\left\{ {1 - {\left\lbrack {\exp\left( \frac{h^{2}\alpha\; t}{k^{2}} \right)} \right\rbrack\left\lbrack {1 - {{erf}\left( \frac{h\sqrt{\alpha\; t}}{k} \right)}} \right\rbrack}} \right\}}}} & (30)\end{matrix}$where the thermal diffusivity (α) is k/ρC_(p), t is the time of exposureto the air stream of velocity V and h is the heat transfer coefficient.A time dependent calculation of this temperature was made for the casewhere the initial face temperate (T_(f)) was 86° F. (30° C.), where inthe calculation of h, T_(a)=−40° F. (−40° C.), V=13.41 mph (21.58Km/hr), P=2116.8 lbf/ft² or 1 atm (sea level conditions) and where thevalues of ρ and C_(p) were the same as those used in the derivation ofEq. (22). With the forced convection coefficient, h=7.16 Btu/hr-ft²-° F.(40.66 W/m²-° C.), for a cylinder in crossflow, the decay in facialtemperature as shown in FIG. 10 was calculated. A computed point (▪) onBuettner's curve represents a drop in the facial temperature (T_(f)) to53.6° F. (12° C.) in 2 min. The single test point (▴) was a measuredfacial temperature noted by Buettner as having been recorded on aparticipant during a parachute drop. Included in FIG. 10 is the facialtemperature decay curve computed from Eq. (22) in which only the forcedconvection heat loss was considered, so that a straightforwardcomparison could be made with Buettner's curve. The difference in thedecay rate of the two curves is believed to be due to the fact thatBuettner assumed a semi-infinite solid, while Eq. (22) was based on acylinder. The fact that Eq. (22) is much closer to the test point (▴)than the Buettner curve demonstrates not only the correctness oftreating the head as a cylindrical surface, but also verifies thecapability of Eq. (22) to correctly predict the facial temperature.

Leblanc et al. Experiment: The second comparison was made with theresults of a study conducted by LeBlanc et al. In this study, each of 25subjects was placed at the end of a wind tunnel blowing air directlyinto their faces at a temperature of −20° C. to 24° C. (−4° F. to 75.2°F.) and at speeds up to 30 mph (48.28 Km/hr). In each case the initialface temperature (T_(fi)) was 30° C. (86° F.). After 10 min. of exposureto these conditions, temperatures were measured on the nose, forehead,and cheek. FIG. 11 shows the decay in the temperatures of the foreheadand cheek as a function of wind velocity for the particular case wherethe air temperature was −20° C. (−4° F.). Also shown in FIG. 11 is thetemperature decay calculated using Eq. (22) assuming only the presenceof radiation and forced convection cooling. The rate of temperaturedecay as predicted by Eq. (22) is nearly identical to that of theforehead, quite possibly because the forehead presents a surface facingthe wind that is more cylindrical than that of the cheek.

Home Freezer Experiments: The third comparison was made with the resultsof three home freezer experiments conducted by the inventors. In all ofthese experiments, different objects were used to simulate the humanhead. Each object was heated to an approximate temperature of the 91° F.to 92° F. (32.78° C. to 33.33° C.), this being the accepted temperatureof the human face when suddenly exposed to a cold environment. Theheated object was then placed in a freezer where the simulated ambienttemperature (T_(a)) varied from −18° F. to 8° F. (−27.78° C. to −13.33°C.). Immediately upon placement in the freezer, the surface temperature(T_(s)) of the object and the freezer temperature were recorded at 5minute intervals. Values of T, were plotted as a function of time (t),curve fitted and then compared with a surface temperature predictionusing Eq. (22). In the absence of any air movement (V=0 mph (0 Km/hr))within the freezer, Eq. (22) predicts the surface temperature only as aresult of a radiation heat loss.

In the first experiment, the head was simulated using a 10 in. (25.40cm) diameter, 2.5 in. (6.35 cm) deep wooden bowl filled with 2.25 lbs(1.02 kg) of tap water heated to a temperature greater than 92° F.(33.33° C.). A thermometer was used to measure the water temperature 5mm (0.0164 ft) below the water surface at the bowl center. When thewater had cooled down to the desired 92° F. to 91° F. (33.33° C. to32.78° C.) temperature range, the bowl was placed in the freezer and thewater temperature (T_(s)) and the freezer temperature (T_(a)) wererecorded in 5 minute intervals over a one hour time period. During thisone hour period, the average freezer temperature was 9° F. (−12.78° C.),while the water surface temperature exhibited the decay shown in FIG.12A. This decay in water temperature and a curve fit of thesetemperatures using the curve fit Eq. (30a) and coefficients of Table 4,below, are in excellent agreement with that predicted by Eq. (22), thusagain confirming the latter's validity.

TABLE 4 Curve fit Eq. (30a) of surface temperature and equationcoefficients for home freezer experiments T_(s)(° F.) = a + b exp(−t/c)(30a) Experiment 1 Experiment 2 Experiment 3 Coefficient Wood BowlChicken Pot Roast a 21.66167 32.845785 36.029454 b 70.427483 57.1333750.681015 c 33.252072 20.349035 14.709987 r² 0.99647891 0.988060310.97708206

The water in the bowl froze at 60 min. after placement in the freezer.In these experiments where only radiation heat losses exist, the closedform solution for the time to freeze (t_(fr)), as shown in Eq. (25c),was used to determine the exposure time at the intersection of the Eq.(22) temperature decay curve with the surface freeze line (T_(s)=32° F.(0° C.)). In this case, t_(fr)=64 min. These results compare favorablywith the actual 60 min. and support the capability of predicting thetime to freeze using the closed form solution of Eq. (25c).

Two other similar experiments were conducted by simulating the headusing food items (chicken, pot roast) that might be found in a homefreezer. FIG. 12B shows the result of using a 5.6 lb (2.54 kg) chickenand FIG. 12C when using a 2.8 lb (1.27 kg) pot roast, each conditionedto a temperature between 92° F. and 91° F. before being placed in thefreezer. Again the surface temperature and the freezer temperature wererecorded at 5 minute intervals over a one hour period. During this onehour period, the average freezer temperatures were 3° F. (−16.11° C.)and −9° F. (−22.78° C.), respectively. These average freezertemperatures, surface temperatures from Eq. (22) and a curve fit ofthese surface temperatures are shown in FIG. 12B and FIG. 12C.

In each of these cases, the agreement with Eq. (22) is not as good as inthe first experiment. The more rapid decrease in surface temperature isprobably due to heat conduction through the outside surface of thesemeat products, whereas in the first experiment this was prevented by theinsulative effect of the wood container. Even though neither of thesecases seem to represent only a radiative heat loss, FIG. 12B does showthat the actual freeze time of 59.5 min. as being very close to thevalue of t_(fr)=57 min. as calculated from Eq. (25c). This may have beenthe case in FIG. 12C also if the intercept point of the temperaturesdecay curve with the freeze line had been determined. Based on thesethree experiments, one may conclude that Eq. (22) likely represents anaccurate expression for the facial temperature decay, irrespective ofthe number of heat processes involved, whereas Eq. (25c) has been shownto provide a good estimate of the time to freeze (t_(fr)) when onlyradiation losses exist.

Discovery Channel™ Experiment: The fourth comparison was made with theresults of an experiment conducted for the Discovery Channel™ (2006) anddescribed in an April 2006 show entitled “I Shouldn't Be Alive: Lost inthe Snow.” A subsequent program entitled “Science of Survival” describedthe experiences of a family who ventured into the wilderness inmid-winter became lost, stranded, and narrowly escaped with their livesin the bitterly cold environment. Because of the family's inexperiencewith survival techniques, the program illustrated the many survivalmeasures that could have been taken to insure their survival.

The “Lost in the Snow” episode emphasized the importance of preventing adrastic drop in the body core temperature (T_(CR)) since a 13° F. dropcan prove fatal. A dexterity test was conducted in a controlled,refrigerated chamber where the subject, Les Stroud a wilderness expert,sat clad only in shorts. The subject ingested a pill with a temperaturesensor to transmit the body core temperature. During the test, thesubject's skin and finger temperatures were monitored, the latter beingimportant at the time when visually the subject might first suffer lossof dexterity.

The ambient temperature (T_(a)) in the chamber was 23° F. (−5° C.). Forthe first 30 min., the experiment was conducted in still air (V=0 mph).At 30 min., an air fan instantaneously increased the air speed to 5 mph(8.05 km/hr) for a period of 6 min. At that point, still air conditionswere resumed. The test was continued at these conditions until thesubject became extremely cold and uncomfortable.

The intent here was to compare the recorded time varying skin and fingertemperatures with the predicted temperature using Eq. (22). There was anaudible difficulty in gathering information off the video recording ofthe program plus the fact that essential information like the initialskin and finger temperatures was not provided. This resulted in anincomplete comparison of the measured and predicted skin temperature.For example, the 10° F. drop in finger temperature over a 15 minuteperiod during the “still air” portion of the test and the measured 43°F. temperature at the end of the 40 minute test period quite clearlyindicated that the initial finger temperature was close to the usual91.4° F. This permitted a comparison of the measured and predictedfinger temperature drop over the entire 40 minute test period. Such wasnot the case for the skin temperature. The first indication of skintemperature is the 81° F. at the end of the 30 minute “still air”portion of the test and its drop to 55° F. at the end of the 6 minuteperiod when the velocity was 5 mph. No information was given on theinitial skin temperature at the start of the “still air” portion. As aresult, the only measured/predicted temperature drop comparison possiblefor the skin was during the 6 minute period.

Before the above comparison was made, it should be noted that one visualobservation was supported by recorded data. The subject is shown to beshivering badly during the test. As discussed in the section METABOLICHEAT GAIN (q_(b)) “Shivering is a thermoregulatory process through whichthe core temperature (T_(CR)) may be restored to its equilibrium orneutral value of 98.2° F. after a lowering.” Since the core temperaturewas shown to remain constant during the test, a lowering of the coretemperature must have taken place before the onset of shivering whichrestored it to its initial value.

The recorded time varying finger temperatures were compared with thetemperature decay curve using Eq. (22) with T_(a)=23° F., V=0 mph,M_(act)=31 Btu/hr-ft² since the subject was involved in minor physicalactivity with his hands and assuming the following: no evaporativecooling ({dot over (w)}=0), no solar heat flux (G=0), sea levelconditions (H=0 ft) and an initial finger temperature (T_(f)) of 91.4°F. This temperature decay curve is shown as the dashed curve of FIG. 13.In “still air”, it was noted from the measured temperatures that thefinger temperature (●) dropped from 60° F. to 50° F. in 15 min.Projecting these points onto the dashed curve of FIG. 13, gives thistime differential as 22.5 min. This larger time differential would becorrect, if and only if, the subject was completely stationary. This isnot quite true. As observed, the subject body may be stationary bysitting in a chair but the fingers were not stationary. The fingermovements were needed to turn knobs and screws in sockets during thedexterity test. The subject's finger speed was estimated based on theobserved movement of the subject's thumb and index finger while turninga screw into a socket. This finger movement speed was estimated to be0.53 in./s (0.03 mph, 0.048 km/hr). With V=0.03 mph, a new temperaturedecay curve was calculated using Eq. (22). This curve lies to the leftof the V=0 mph curve. Projecting the finger temperature (●) to the leftonto this curve, they (∘) are now 10 min. apart. Since the measured 10°F. temperature differential actually took place during the first 30 min.of the test, this curve shift to the left clearly demonstrates that thefingers were in motion. Because this temperature decay curve terminatedat 30 min., a follow on section of the curve between 30 and 36 min. wascalculated using Eq. (22) with an initial finger temperature (T_(f)) of50° F. and V=5 mph. The final section of the curve between 36 min. and40 min. was again calculated using Eq. (22) with the initial fingertemperature of 37° F. at 36 min. and V=0.03 mph. The final fingertemperature at 40 min. was found to be 35° F. Because Eq. (22)determined that the 10° F. temperature drop occurred over 10 min. ratherthan the actual 15 min. and because the final finger temperature wasfound to be 35° F. and not the actual 43° F., the predictions can onlybe considered fair. The lack of better agreement is attributed to aprobable and unaccountable heat transfer between the subject's hands andthe devices that he was attempting to manipulate.

The third curve of FIG. 13 is the result of applying Eq. (22) todetermine the skin temperature (▪) drop from 81° F. over the 6 minuteperiod when V=5 mph. The predicted drop of 34° F. from 81° F. to 47° F.is again only a fair comparison with the actual 26° F. drop from 81° F.to 55° F. This difference between the predicted (34° F.) and themeasured (26° F.) temperature drops may be explained by the fact thatthe skin temperature was being sensed at the external oblique of theabdomen which, being sheltered from the 5 mph wind via a table in frontof the subject, would result in a smaller temperature drop.

Adamenko and Khairullin Experiment: Having established the validity ofEq. (22), it was then used to calculate values of T_(f) after 30 min.for a comparison with the Adamenko and Khairullin values of T_(f) fromEq. (29). In doing so, the values of H, M_(act), {dot over (w)}, and Gused in Eq. (22) had to be specified. Schwerdt notes that the Adamenkoand Khairullin experiment was conducted in four Russian cities each ator near sea level. Thus, H was set equal to zero. The forty peopleinstrumented were instructed to walk slowly during the test when thetemperature on each of the facial components was recorded. Walkingslowly suggests that the participants in the test were engaged inminimum physical activity. Therefore, the associated metabolic heatgeneration may be that for “walking about” as shown in Table 1, above,in which case M_(act)=31 Btu/hr-ft² (97.79 W/m²) was used in thecalculation of the metabolic heat flow. For minimum physical activity,{dot over (w)}=0.00655 lbm/hr-ft² (0.032 kg/hr-m²) was used. The valueof the solar radiation (G) during the test is unknown, although Adamenkoand Khairullin stated that “[s]ufficiently intense solar short-waveradiation was also taken into account.”

Since the value of G during their experiment is unknown, the approachtaken here was to determine if an assumed value of G in Eq. (22) would,at a given ambient temperature (T_(a)) and a velocity (V), yield valuesof T_(f) at 30 min. that are in reasonably good agreement with Adamenkoand Khairullin values from Eq. (29). This approach was restricted by thefact that the selected value of G could not exceed the maximum valueexpected for the four Russian cities where the tests were conducted.Because the latitude at the Russian test sites was higher than that ofthe northernmost region (45°-50°) in the US, where G=37.78 Btu/hr-ft²(119.17 W/m²) in Table 2, the maximum value of G was estimated to be32.0 Btu/hr-ft² (100.95 W/m²). Results of a calculation with an assumedvalue of G=5 Btu/hr-ft² (15.77 W/m²) are shown in FIG. 14A for theparticular case where T_(a)=0° F. (−17.78° C.), V=0, 20 and 40 mph (0,32.19 and 64.37 Km/hr). Included in FIG. 14A are the Adamenko andKhairullin results (▪) from Eq. (29) shown plotted at the 30 minuteexposure time. The immediate reaction when one views this figure is thatthere is no agreement between the two sets of results at the two highervelocities (20 mph, 40 mph (32.19 Km/hr, 64.37 Km/hr)). Schwerdt, incommenting on the Adamenko and Khairullin results, stated that at thehigher wind speeds the skin will freeze “closer to a half hour than anhour”. This is shown in FIG. 14A where the two Adamenko and Khairullinvalues (▪) for 20 mph and 40 mph (32.19 Km/hr and 64.37 Km/hr) liefairly close to the freeze line (T_(f)=32° F. (0° C.) at the 30 minuteexposure time. But, this is far different from the results using Eq.(22), which shows freeze times of less than 4 min. for both of thesespeeds. The very rapid increase in the rate of decay of the facialtemperature as velocity is increased from 0 mph to 40 mph (0 Km/hr to64.37 Km/hr) suggests the possibility that the Adamenko and Khairullinmeasurements failed to detect this sensitivity to velocity. Perhaps thisled to their conclusions that the freeze times at these two highervelocities would be 30 min. rather than the more exact time of less than4 min. On the other hand, Schwerdt noted that at the lower speeds, V=0mph (0 Km/hr) in the present case, the Adamenko and Khairullin resultswould predict freezing closer to the one hour period rather than the onehalf hour period. This is clearly demonstrated in FIG. 14A by the factthat the V=0 mph curve not only passes close to the V=0 mph (▪) point att=30 min., but actually passes exactly through the V=0 mph (□) point att=60 min. This would seem to confirm Schwerdt's statement of “closer toone hour than one half hour” and would seem to prove at least a partialagreement with these test results. However, G=5 Btu/hr-ft² (15.77 W/m²)would not seem to reflect the “intense solar short-wave radiation” thatexisted during the tests. Perhaps it represents a greatly reduced solarradiation on a cloudy day as opposed to the assumed maximum solarradiation on a clear day. Intense solar radiation would logically seemto be a value near the assumed maximum value of 32 Btu/hr-ft² (100.95W/m²). But if G were increased from 5 to 32 Btu/hr-ft² (15.77 to 100.95W/m²), the V=0 mph curve would be displaced increasingly upward and awayfrom the freeze line (T_(f)=32° F.).

FIG. 14B, with G=7.8 Btu/hr-ft² (24.61 W/m²), represents what isprobably the near maximum value of G, whereby the resulting freeze timeof 79 min. can still be called close to the V=0 mph (□) point at t=60min. Since this slight increase in G from 5 to 7.8 Btu/hr-ft² left thedifferences between the two results virtually unchanged at the twohigher velocities, one concludes that if any agreement exists with theAdamenko and Khairullin test results concerning the exposure time, it isonly at calm (V=0 mph) conditions and then only if the tests wereconducted under overcast conditions, or if sunshine was intermittent andnot continuous during the tests.

With Eq. (22) validated, it can be combined with Eq. (18a) to determinethe time variation in the wind chill temperature experienced by anindividual during an extended exposure. Consider an individual that isexposed to ambient conditions of T_(a)=0° F. (−17.78° C.) and V=20 mph(32.19 Km/hr). Assume the absence of sunshine (q_(i)=0). Realistically,the evaporative heat loss (q_(e)) and the metabolic heat gain (q_(b))are always present. In view of making a cursory comparison with theBluestein and Zecher results, it was also assumed that q_(e)=q_(b)=0.

FIG. 15 shows the decay in facial temperatures (T_(f)) with exposuretime as calculated from Eq. (22). At selected facial temperatures alongthis curve, the wind chill temperature (T_(wc)) was calculated from Eq.(18a). The results of FIG. 15 show a very significant warming of thewind chill temperature as the facial temperature decreases. Bydefinition, the exposure time when the facial temperature has decreasedto 32° F. (0° C.) is called the time to freeze (t_(f)). In this case,where t_(f)=3.67 min., the wind chill temperature has increased, that ismoderated, from its initial value of −54.16° F. to −40.20° F. (−47.87°C. to −40.11° C.) for an increase of 13.96° F. (7.76° C.). Ananticipated moderation such as this in the Siple and Passel values ofthe wind chill temperature seems to have been the motive behind thedevelopment of the Bluestein and Zecher model. Unfortunately, theBluestein and Zecher wind chill model failed to show any measurableamount of moderations as indicated by the comparison in FIG. 8.

Time to Freeze: In addition to providing a more precise prediction ofthe wind chill temperature (T_(wc)), a key feature of the wind chillmodel of the present invention is its capability of determining theexposure time when facial freezing will occur. The wind chilltemperature, although a true sensed temperature, becomes subjective innature, since individuals may differ greatly in their discernment of itsactual magnitude. For example, an individual is probably more likely todetect the difference between two warmer wind chill temperatures like30° F. (−1.11° C.) and 15° F. (−9.44° C.) than two colder andpotentially more dangerous ones like −20° F. (−28.89° C.) and −40° F.(−40° C.). When the time to freeze (t_(f)) is specified along with thewind chill temperature, it frees the individual from having to beconcerned with the actual value of the temperature. Instead it becomes awarning to the individual of a limiting time after exposure when facialfreezing will occur. It has the added benefit of allowing the individualto preplan outdoor activities so as not to exceed this time limit. Thiswas the basis for developing the time to freeze expression of Eq. (24).

There is no known experimental data or any existing equation for facialfreezing time against which the predictions of Eq. (24) can be compared.However, actions taken by some contestants in the World Downhill SkiChampionships during the 2003/2004 racing season confirmed the validityof Eq. (24) through the use of Eq. (22), from which it was derived.These races were conducted in the European Alps at what was considered acolder than normal temperature (T_(a)), believed to have been less than10° F. (−12.22° C.). On this course, the contestants reached a velocity(V) of 80 mph (128.74 Km/hr) in about 6 seconds after leaving thestarting gate. The total time to run the course was expected to be justunder two min. The local racers, who were familiar with the course, wereapparently aware from past experiences of the possibility of facialfreezing under these conditions (T_(a), V). As a result, these racerstook precautions to prevent or minimize freezing by applying tape stripsto their faces. At the finish line, none of the racers appeared to haveexperienced frostbite. The warming effect of sunshine could not havebeen a factor in the absence of frostbite, since the race course was inthe shadow cast by the mountain. Altitude, which will be shown later tohave the effect of delaying facial freezing, may have had some effect.The average altitude was estimated to be 8,000 ft (2,438.43 m) based onan estimated altitude of 10,000 ft (3,048.04 m) at the starting gate anda 4,000 ft (1,219.21 m) elevation drop to the finish line. The verystrenuous activity of racing downhill at a speed of 80 mph wouldgenerate a higher than normal metabolic heat flow rate to the face, andthat might have had a slight effect on forestalling frostbite.

Table 1, above, shows the normal heat generation (M_(act)) for downhillskiing as 96 Btu/hr-ft² (302.84 W/m²) and that for cross-country skiingas 83 Btu/hr-ft² (261.83 W/m²). The average velocity of a non-racingdownhill skier might be 25 mph (40.23 Km/hr), whereas that for across-country skier might be 3 mph (4.83 Km/hr). It is postulated thatthe metabolic heat generation varies as the square of the velocity, sothat 2 VdV/dM_(act)=[2(14)(25−3]/(96−83)=47.38 (mph)²(hr-ft²/Btu)=3.669×10⁵ ft⁴/Btu-s (3 m⁴/J-s). With this gradient, themetabolic heat generation for the downhill racer would be M_(act)=217.89Btu/hr-ft² (687.35 W/m²). This value is much larger than that (96Btu/hr-ft² (302.84 W/m²)) for the more casual, non-racing downhill skierbecause of the much greater muscle exertion required to executedirection changing at the higher speed. Finally using the guidelinespreviously set forth, the racers' evaporation flux rate would be {dotover (w)}=0.02183 lbm/hr-ft² (0.1066 kg/hr-m²), which, although amaximum, most certainly did little to enhance the possibility offrostbite.

None of the racers experienced facial freezing upon completion of therace at about 1 minute 50 seconds (1.83 min.), including the locals whoanticipated this possibility. FIG. 16 shows the results of calculatingthe facial temperature decay curves for the ski racers using Eq. (22)and the above set of conditions: V=80 mph, −5° F.<T_(a)<10° F. (−20.56°C.<T_(a)<−12.22° C.), G=0 Btu/hr-ft², H=8,000 ft (2,438.43 m),M_(act)=217.89 Btu/hr-ft² and {dot over (w)}=0.0283 lbm/hr-ft². Theresults show that at the finish line, at t=1.83 min., none of the racerswould have been expected to experience facial freezing over thistemperature range, and none did. The belief of the local racers thatfreezing could have occurred was justified. For example, if the ambienttemperature had been T_(a)=0° F. (−17.78° C.) and if a racer hadexperienced some misfortune on the course, causing his arrival at thefinish line to be 19 seconds later at 2.15 min., he would haveexperienced facial freezing. But a time increment of 19 seconds is notwhat would be expected in a race of this nature from a group ofworld-class racers. Now if the temperature had been T_(a)=−5° F.(−20.56° C.), then the freezing time would be t_(f)=1.98 min., which isonly 0.15 min. or 9 seconds past the nominal completion time (1.83min.). A 9 second variation in a competitor's completion time isentirely possible, and could have been the basis for the concernexpressed by the local racers. Thus, the actual ambient temperature mayhave been close to −5° F. (−20.56° C.). This example furtherdemonstrates the usefulness and the apparent accuracy of Eq. (22) tocalculate the facial temperature (T_(f)) and consequently Eq. (24) tocalculate the time to freeze (t_(f)).

Having verified Eq. (24), a prediction of the time to freeze using thisequation was compared against two existing approximations. A case inpoint was the pre-dawn weather forecast for the morning of Jan. 5, 2004in Minneapolis, Minn. The weather conditions were stated as T_(a)=−4° F.(−20° C.), V=13 mph (20.92 Km/hr) with a T_(wc)=−23° F. (−30.56° C.),the latter correctly being the value of wind chill temperature from thecurrent wind chill model. Intended to warn people walking to and waitingat bus stops during the dark morning hours of the potential severity ofthis T_(wc), the comment was made that under these conditions “freezingwould occur in 30 min.”. This comment suggests the meteorologist'sfamiliarity with an equation like Eq. (29) which, as previouslydiscussed, expresses facial temperature (T_(f)) after 30 min. ofexposure to a given set of ambient conditions (T_(a), V). Because theabove values of T_(a)=−4° F. and V=13 mph in Eq. (29) result in a facialtemperature of exactly 32° F. (0° C.) after 30 min., the comment waslikely made to warn the listeners of the possibility of facial freezing,should their exposure time equal or exceed 30 min. Unfortunately, thiswas misleading advice for two reasons. First, Eq. (29) was derived fromthe Adamenko and Khairullin test results where their experiment wasconducted under partially sunny conditions. Consequently, Eq. (29) wouldreflect an increase in the facial freezing time (t_(f)) as a result ofsolar radiation. Since sunshine was absent during the dark morninghours, a freezing time of 30 min. would be expected to be anover-prediction. Secondly, as the results of FIG. 14 have shown, theAdamenko and Khairullin results (▪) overpredict the time to freeze asvelocity increases from the calm (V=0 mph) condition. Therefore, at V=13mph the time to freeze would be expected to be much less than 30 min. Infact, this was the case, because the actual freezing time in theMinneapolis forecast based upon computed values using Eq. (24) and shownin FIG. 17 was t_(f)=4.15 min. This statement of a possible 30 minutefreeze time was obviously made with good intentions of warning thelistening public. However, incorrectly stating a possible freezing timethat is so much larger places the public at greater risk by minimizingthe urgency to avoid critical exposure.

Aside from using Eq. (29) to determine the freezing time, theMinneapolis meteorologist could have used an equation developed byEnvironment Canada (2002) to approximate the “min. to frostbite” for 5%of the population that is most susceptible to frostbite. The equationis,t _(f)={[−24.5(0.667V+4.8)]+2111}(−4.8−T _(a))^(−1.668)  (31a)where T_(a) is in ° C. and V is in km/hr. The equation is limited to−50° C.≦T_(a)≦15° C. (−58° F.≦T_(a)≦59° F.) and 20 km/hr≦V≦80 km/hr(12.43 mph≦V≦49.71 mph). Expressed in English units, the equation is,t _(f)={[−24.5(1.067V+4.8)]+2111}(12.978−0.556T _(a))^(−1.668)  (31b)where T_(a) is in ° F. and V is in mph. With the weather forecastconditions of −4° F. and V=13 mph, Eq. (31b) predicts the time to freezeas t_(f)=17.70 min. Again an approximate value greatly over predicts thetime to freeze. This coupled with the fact that it does not seem toapply to the remaining 95% of the populace makes any application of Eq.(31b) pointless. The above applications clearly show that neither of thecurrently available approximations, Eq. (29) and Eq. (31b), can be usedto predict the onset of facial freezing.

Since neither of the two above expressions can correctly predict thefreezing time even when applied within their specific ranges (T_(a), V)of applicability, they certainly cannot be expected to predict the timeto freeze when either the ambient temperature or wind velocity or bothlie outside of these ranges. An extreme example of this is the earth'scoldest recorded temperature of T_(a)=−128° F. (−88.89° C.) inAntarctica. Only Eq. (24) permits a calculation of the exact time tofreeze (t_(f)) for personnel located there who may have been exposed tothis temperature. It is not known with certainty whether windy or calmconditions prevailed. The assumption was made that there was no windreduction (WRF=0) and that this temperature occurred during hours ofdarkness (G=0 Btu/hr-ft² (W/m²)). Also the altitude would have been nearsea level (H=0 ft (0 m)). Calculation of the time to freeze (t_(f)) forindividuals who may have been exposed to this severe temperature wasmade for two velocities V=1 mph (1.61 Km/hr) and V=4 mph (6.44 Km/hr).From Eq. (24) the time to freeze when T_(a)=−128° F. (−88.89° C.) wast_(f)=4.45 min. when V=1 mph and t_(f)=2.53 min. when V=4 mph. These arevery low values of t_(f) which could not have been accurately determinedby either of the above approximate expressions.

To better visualize the effect of velocity, the values of t_(f) werealso calculated using Eq. (22) to determine these values from theintercept point of the facial temperature decay curves with the freezeline (T_(f)=32° F. 0° C.)). These results are shown in FIG. 18 for anindividual “standing relaxed”, i.e., from Table 1, M_(act)=22 Btu/hr-ft²(69.4 W/m²) and therefore w=0.00655 lbm/hr-ft² (0.032 kg/hr-m²).Inclusion of q_(b) and q_(e) in the calculations was not reallynecessary since earlier results had shown that their effects on T_(f)were negligible. In FIG. 18, the values of T_(f) for V=1 mph and 4 mphare 4.36 and 2.44 min., respectively. These are close to the abovepredicted values from Eq. (24). FIG. 18 shows that for calm conditions(V=0 mph), the value of t_(f)=22.2 min. is much higher than thet_(f)=4.36 min. when V=1 mph. This demonstrates again the powerfuleffect of the forced convection coefficient on reducing the time tofreeze. Since a minor increase in wind velocity causes such a majordecrease in t_(f), the question arises concerning what effect on t_(f)the most extreme wind condition would have. Assuming a maximum windspeed of V=80 mph (128.74 Km/hr), from Eq. (22) and approximately so,from FIG. 18, the minimum time to freeze that could possibly ever existwas t_(f)=0.63 min. or 38 seconds.

One of the objectives of the model developed here was to define thefacial freezing time corresponding to a given wind chill temperature.This time should assist a user to “quantify” the value of the wind chilltemperature, particularly at its extreme lower values where sensing itbecomes very subjective. However, as already demonstrated, unless theannounced freezing time is accurate, the listening public may experiencemore harm than good as a result of it. It is the inventors' belief thatonly Eq. (24) provides this accuracy, since it was based on atheoretical development of the facial temperature, which, in turn, hasbeen substantiated by experimental data. It should be noted that Eq.(24) along with its constants in Table 4 were derived for sea levelconditions (H=0 ft (0 m)). However, the time to freeze is altitudedependent. This will be discussed below.

Effects of Heat Losses/Heat Gains on the Wind Chill Temperature and Timeto Freeze: Each of the three heat losses (q_(fc), q_(r), q_(e)) and eachof the two heat gains (q_(i), q_(b)) were individually examined todetermine their contributions to the wind chill temperature and the timeto freeze.

Wind Chill Temperature: Eq. (18a) was used to calculate the effect ofeach heat loss and heat gain on the wind chill temperature (T_(wc)).Calculations were made for an assumed Boston, Mass. resident casuallywalking on a sunny, cold winter day when the ambient temperature (T_(a))is 0° F. (−17.78° C.). Boston is approximately at sea level and is inthe mid latitude region, H=0 ft (0 m) and G=42.66 Btu/hr-ft² (134.58W/m²) from Table 2. From Table 1 the metabolic heat generation for“walking about” is M_(act)=31 Btu/hr-ft² (97.79 W/m²) and therefore thewater evaporation flux rate becomes {dot over (w)}=0.00655 lbm/hr-ft²(0.032 kg/hr-m²). It was assumed that the initial facial temperature(T_(f)) was 91.4° F. (33° C.) and that there was no wind speed reductionat head level, thus WRF=0.

FIGS. 19A-B show the effect of the heat losses on the wind chilltemperature. The top curve of FIGS. 19A-B shows the wind chilltemperature due to q_(fc) alone. FIG. 19B is an enlarged view of theregion shown in the dotted hexagon in FIG. 19A. By definition, a windchill temperature is less than the ambient temperature. If q_(fc) wasthe only heat loss present, the resident would not be experiencing awind chill temperature until the wind speed (V) exceeds 6 mph (9.66Km/hr). At an arbitrary velocity of V=21 mph (33.8 Km/hr), he wouldexperience a wind chill temperature T_(wc) of −48.7° F. (−44.83° C.).However, q_(fc) will never be the only heat loss. Rather a radiation(q_(r)) heat loss will always be present as long as the facialtemperature (T_(f)) is cooling down towards the ambient temperature(T_(a)). Therefore, adding the ever present radiation heat loss to theforced convective heat loss, q_(fc)+q_(r), decreases the T_(wc)uniformly over the entire velocity range. Now the resident willexperience wind chill when V exceeds 4 mph (6.44 Km/hr) and at velocityof V=21 mph (33.8 Km/hr), this decrease in T_(wc) due to the addition ofthe radiation heat loss is −7.6° F. (4.22° C.) from −48.7° F. (−44.83°C.) to −56.3° F. (−49.06° C.). This q_(fc)+q_(r) curve corresponds tothe Siple and Passel values (T_(wc)=−39.2° F. (−39.56° C.)) at T_(a)=0°F., except that it shows lower values of T_(wc) because of thepreviously discussed difference in the forced convection coefficients.The addition of the evaporative heat loss as shown in the lowest curve,q_(fc)+q_(r)+q_(e), results in an additional 0.8° F. (0.44° C.) decreasein T_(wc) from −56.3° F. (−49.06° C.) to −57.1° F. (−49.50° C.). Thisdecrease in T_(wc) due to q_(e) is only 10.5% of that due to q_(r). Thiscurve, q_(fc)+q_(r)+q_(e), which is the sum of the three losses,represents the lowest possible value of the wind chill temperature.

FIGS. 20A-B show the effect of heat gains on the lowest possible windchill temperature curve, q_(fc)+q_(r)+q_(e), from FIG. 19B. FIG. 20B isan enlargement of the region shown as a dotted hexagon in FIG. 20A. Nowthe moderating effect on T_(wc) can be determined by subtracting out theeffect of the heat gains q_(b) and q_(i). Subtracting out q_(b) first,as in the middle curve, q_(fc)+q_(r)+q_(e)−q_(b), shows that at V=21 mphthere is only a 0.5° F. (0.28° C.) warming effect of the metabolic heaton the T_(wc) to raise it from −57.1° F. (−49.50° C.) to −56.6° F.(−49.22° C.). This 0.5° F. increase in T_(wc) due to q_(b) is 6.6% ofthe decrease (7.6° F.) due to q_(r) and brings T_(wc) back to a valuethat is 0.3° F. (0.17° C.) lower than the previous value of −56.3° F.(−49.1° C.) in FIG. 19B, before it was decreased by q_(e). Thus, q_(e)and q_(b) have small and nearly equal offsetting effects on T_(wc).Finally, subtracting out the solar heat gain (q_(i)) as shown in theupper curve, q_(fc)+q_(r)+q_(e)−q_(b)−q_(i), increases the T_(wc) by2.7° F. (1.5° C.) from −56.6° F. (−49.22° C.) to its final value of−53.5° F. (−47.5° C.). This 3.1° F. increase is 40.8% of the decrease(7.6° F.) in T_(wc) due to q_(r). The 3.6° F. (1.78° C.) differencebetween the coldest value of T_(wc) of −57.1° F. and the warmest valueof −53.5° F. may not matter to the individual, because at these very lowtemperatures he is unlikely to detect the difference.

Several comments and conclusions can be drawn from FIGS. 19A-B and FIGS.20A-B concerning the effects of heat losses and heat gains on the windchill temperature (T_(wc)). First, radiation heat loss (q_(r)), notforced convection heat loss (q_(fc)), is the key component in theT_(wc). It will always exist as long as there is a temperaturedifferential between the face and the ambient temperature. The forcedconvection heat loss will be nonexistent in the absence of wind and theevaporative (q_(e)) and metabolic (q_(b)) terms, if they do exist, willhave small and counteracting effects.

Second, the radiation heat loss (q_(r)) is at a maximum at the moment ofexposure when the facial temperature T_(f) is 91.4° F. (33° C.). Onlywhen T_(f) decreases in time and approaches the ambient temperature(T_(a)) will the radiation heat loss decrease to the point that theforced convection heat loss (q_(fc)) may dominate.

Third, the effect of the evaporation heat loss (q_(e)) on T_(wc) isnearly offset by the metabolic heat (q_(b)) gain. This might have beenexpected since both depend upon the level (M_(act)) of physicalactivity. Although this was shown to be true for the low level activityof “walking about” considered here, it is likely to be true for allactivity levels. From this it can be concluded that (q_(e)) and (q_(b))can probably be safely neglected if only an approximation of T_(wc) isdesired.

Fourth, the effect of the solar heat gains (q_(i)) on T_(wc) is muchgreater than that of (q_(e)) and (q_(b)). Since q_(i) is constant forany geographical location but q_(r) decreases as T_(f) decreases withincreasing exposure time, it is possible that after an extended periodof time the solar heat gain may completely offset the radiation heatloss.

Fifth, the negligible effect of q_(e) and q_(b) on T_(wc) during ambientconditions, when wind chill is of concern, would seem to contradict thestatement made by some investigators that physical activity will have amoderating or warming effect on the wind chill sensed by an individual.Actually, there may not be a contradiction, since two differentphenomena are being considered here: one, the heat loss from a fullyclothed body, the other, the heat loss from the exposed face. For thefully clothed individual participating in a physical activity, it mightbe reasoned that the potential heat loss (q_(e)) due to perspirationover the clothed portion of the body may not exist if the sweat isprimarily being absorbed and retained by the clothing. In addition, themetabolic heat (q_(b)) flowing towards the skin surface is probablyaccumulating in the sense that it is largely prevented from leaving theskin surface due to the insulating effect of the clothing. This additiveeffect of q_(e) and q_(b) would result in an overall warming effectwhich would raise the individual's comfort level as claimed. What mustbe considered here is that this warming effect in no way represents amoderation of the wind chill temperature (T_(wc)) sensed by theindividual, since this temperature is sensed within the skin's dermislayer of the exposed face and not within the clothed body. Thus,physical activity can help an individual feel warmer in a wind chillingenvironment without really affecting the wind chill temperature.

Time to Freeze: Eq. (22), rather than Eq. (24), was used to calculatethe effect of each heat loss and gain on the time to freeze (t_(f)).Calculations were made for the same set of conditions used to obtain theeffects of heat losses and gains on the wind chill temperature (T_(wc))as shown in FIGS. 19A-B and FIGS. 20A-B for a Boston resident walking ona sunny, cold winter day when the ambient temperature (T_(a)) was 0° F.(−17.78° C.). Again, the following conditions applied: H=0 ft (0 m),G=42.66 Btu/hr-ft² (134.58 W/m²), M_(act)=31 Btu/hr-ft² (97.79 W/m²),{dot over (w)}=0.00655 lbm/hr-ft² (0.032 kg/hr-m²), WRF=0 and theinitial facial temperature was 91.4° F. (33° C.). Eq. (22) was used todetermine the facial temperature decay and the times to freeze (t_(f))for different combinations of heat losses and gains. Calculations weremade at V=0 mph (0 Km/hr) and V=40 mph (64.37 Km/hr) and at ambienttemperatures of T_(a)=0° F. (−17.78° C.) and −40° F. (−40° C.). Theresults are shown in FIGS. 21-23.

FIG. 21A, with T_(a)=0° F. and V=40 mph, shows the effect of heat lossesand heat gains in various combinations on the facial temperature decay,and particularly on the time to freeze (t_(f)). FIG. 21B shows anenlargement of the region shown in the dotted hexagon illustrated inFIG. 21A. First, only the effect of forced convection (q_(fc)) wasconsidered. Though realistically unobtainable by itself in a realsituation, it provided a means of checking the accuracy of the forcedconvection component in Eq. (22). In FIG. 21B, when only q_(fc) isconsidered in Eq. (22), the time to freeze is t_(f)=2.94 min.Calculating the time to freeze from the closed form solution in Eq.(26c) gives almost the same value t_(fc)=2.95 min. This excellentagreement confirms the accuracy of the forced convection component inEq. (22). Next, a combination of the two major heat losses,q_(fc)+q_(r), in FIG. 21B gives the time to freeze as t_(f)=2.684 min.Adding the evaporative heat loss (q_(e)) decreases the t_(f) to itsminimum value of 2.644 min. This is the minimum value of t_(f).Similarly, the addition of q_(e) to q_(fc)+q_(r) in FIGS. 19A-B resultedin the minimum (coldest) wind chill temperature (T_(wc)). The decreasein t_(f) due to the addition of q_(e) was 0.04 min. in going from 2.684min. to 2.644 min. Now subtracting out the metabolic heat gain (q_(b))from the sum of the three heat losses (q_(fc), q_(r), q_(e)) increasesthe time to freeze by 0.021 min. in going from 2.644 min. back up to2.665 min., which is slightly less than the value of t_(f) (2.684 min.)before q_(e) and q_(b) were considered. Because they essentially canceleach other, it appears that t_(f) can be safely predicted by neglectingthe effects of q_(e) and q_(b). This is true for the minimum level ofactivity (“walking about”) considered here. Since both q_(e) and q_(b)increase with increasing level of activity, the cancellation may applyat all times irrespective of the activity level. However, this cannot besaid with certainty, since the exact relationship of {dot over (w)} toM_(act) is not known. What can be said with certainty is that at theseconditions of high velocity (V=40 mph) and moderately cold temperature(T_(a)=0° F.), a low level of physical activity like walking is notbeneficial to the individual, in that there is a decrease and not anincrease in t_(f). This may not be true for higher levels of activity.Finally, subtracting out the solar heat gain (q_(i)), which isequivalent to adding the presence of sunshine, decreases the rate offacial cooling and therefore increases t_(f). This means that theindividual being considered here, because he/she is walking in sunshine,will encounter facial freezing at t_(f)=2.855 min. and not t_(f)=2.665min. in the absence of sunshine. This 0.19 min. increase in time tofreeze in going from 2.665 min. to 2.855 min., although not substantialin this case, does indicate a beneficial effect of sunshine. For theambient conditions considered above, when sunshine is present, thefacial freezing time is calculated to be 2.81 min., 2.855 min., and 2.90min. for thermal absorptivities of 0.5, 00.65, and 0.8, respectively.The corresponding maximum change in the calculated facial freezing timeis approximately 3%.

FIG. 22A with T_(a)=−40° F. (−40° C.) and V=40 mph (64.37 Km/hr) showsthat decreasing the ambient temperature from 0° F. to −40° F. whilemaintaining V=40 mph decreases the time to freeze (t_(f)) by more than40% when all heat losses and heat gains are present. FIG. 22B shows anenlargement of the region outlined in the dotted hexagon in FIG. 22A.The decrease in the time to freeze (t_(f)) is shown by the decrease int_(f) from 2.855 min. for T_(a)=0° F. in FIG. 21A-B to t_(f)=1.621 min.when T_(a)=−40° F. This decrease in ambient temperature from 0° F. to−40° F. decreases the reduction in t_(f) due to q_(e) from 0.040 min. to0.011 min. It also decreases the increase in t_(f) due to q_(b) from0.021 min. to 0.006 min. The result is that the counteracting effects ofq_(e) and q_(b) on t_(f) decrease even further with decreasing ambienttemperature. Again, there is no beneficial effect to the individual interms of an increase in t_(f). However, the decrease in t_(f) due tothis low level of activity (walking) is much less. Unless higher levelsof activity prove otherwise, physical activity can be neglected in thedetermination of t_(f) at the lower ambient temperatures and higher windvelocities when freezing is most likely to occur. At this much lowerambient temperature, an already minor beneficial effect of sunshine isalso reduced. Without sunshine, t_(f)=1.571 min. and with sunshine,t_(f)=1.621 min.; this is an increase of 0.05 min. compared to 0.19 min.when T_(a)=0° F. Similarly, for the ambient conditions considered above,when sunshine is present, the facial freezing time is calculated to be1.61 min., 1.621 min., and 1.633 min for thermal absorptivities of 0.5,0.65, and 0.8, respectively. The corresponding maximum change in thecalculated facial freezing time is approximately 2%.

FIG. 23A and enlargement shown in FIG. 23B with T_(a)=0° F. and V=0 mph(0 Km/hr) show that decreasing the velocity from the V=40 mph in FIGS.21A-B results in large increases in the time to freeze (t_(f)) that is16 to 20 times greater than those in FIGS. 21A-B. This is because theforced convection heat loss (q_(fc)) is nonexistent when V=0 mph.Consequently, its effect of causing a rapid decay in the facialtemperature, as evidenced by the results of FIG. 14, is no longerpresent. The absence of the q_(fc) heat loss causes the facialtemperature to decrease more gradually so that the facial temperaturedecay curve intercepts the freezing line (T_(f)=32° F. (0° C.)) at amuch larger exposure time. This absence of q_(fc) is represented by thehorizontal line which signifies that it has no effect on the facialtemperature decay, leaving the radiation heat loss (q_(r)) as the onlymajor remaining term. Eq. (22) predicts the time to freeze for thisradiation term as t_(f)=53.8 min. From the closed form solution of Eq.(25c), the time to freeze is t_(fr)=53.8 min. In this case, theexcellent agreement of these two numbers now confirms the accuracy ofthe radiation component in Eq. (22). The addition of the evaporationcomponent (q_(e)) reduces the time to freeze by 12.62 min. tot_(f)=41.55 min., while the subsequent subtraction of the metaboliccomponent (q_(b)) increases the time to freeze by 5.65 min. back up tot_(f)=47.20 min. This shows that at T_(a)=0° F. and a calm condition(V=0 mph), the effects of q_(e) and q_(b) no longer nearly cancel eachother as they did in FIGS. 22A-B for the lower ambient temperature andthe higher wind velocity where facial freezing is likely to occur.Although the difference between these components (q_(e), q_(b)) is muchgreater at the lower temperature (T_(a)=0° F.) and a calm (V=0 mph)condition, there still is no beneficial effect of physical activity fromthe standpoint of an increase in the time to freeze. Based on this andthe above results, one concludes that it is to an individual's benefitto refrain from any low level of physical activity if he desires toavoid a decrease in the time to freeze. The possibility exists that amore energetic form of physical activity may increase the t_(f).Finally, subtracting out the solar radiation (q_(i)) results in atemperature decay curve shown as q_(r)+q_(e)−q_(b)−q_(i) that does notintersect the freezing line (T_(f)=32° F. (0° C.)). This means that inthe presence of sunshine, facial freezing will not take place at theseambient conditions. This emphasizes the powerful effect of sunshineunder calm (V=0 mph) conditions.

Finally, FIG. 24 is a plot of the facial temperature (T_(f)) curves whenT_(a)=0° F. (−17.78° C.) using the infinite series expression of Eq.(23h). Note that the initial facial temperature in this case is the coretemperature (T_(CR)) of 98.2° F. (36.78° C.) and not 91.4° F. (33° C.).Since the infinite series expression applies only when a forcedconvection heat loss takes place at the facial surface, it compares withthe results of the lumped capacitance approach shown in FIGS. 21A-Bwhere only the forced convection (q_(fc)) was being considered. As shownin FIGS. 21A-B when T_(a)=0° F. and V=40 mph, t_(f)=2.94 min. using Eq.(22). At these same ambient conditions, t_(f)=3.18 min. from FIG. 24.This small difference of 0.24 min. (14.4 seconds) between these twovalues not only validates Eq. (22), but also validates the lumpedcapacitance approach used in its derivation. The comparison is evenbetter when it is noted that the initial mid-segment temperature in thisinfinite series method is 98.2° F., while in FIGS. 21A-B, the initialtemperature is 91.4° F. A linear temperature correction on t_(f) showsthat its actual value in this method is t_(f)=91.4/98.2×(3.18)=2.96 min.This smaller difference of 0.02 min. (1.2 s) between this value andt_(f)=2.94 min. is a further validation of Eq. (22) and its derivation.The infinite series solution of Eq. (23h) is limited to only “forced”convection conditions at the facial surface. In real life, where othermodes of heat transfer will almost certainly exist, the lump capacitancemethod of Eq. (22) provides the only means of determining t_(f). Theadded benefit of the lump capacitance method of Eq. (22) is that it ismore general, simpler, and easier to use.

Several comments and conclusions can be made concerning the effect ofheat losses and gains on the time to freeze (t_(f)). First, the effectof the heat loss due to evaporation (q_(e)) is nearly offset by the heatgain due to body metabolism (q_(b)) at the lower ambient temperaturesand higher velocities when the danger of facial freezing is thegreatest. This is true for a low level of physical activity and may betrue for higher levels as well. At low levels of physical activity, likewalking, an individual will experience a decrease rather than anincrease in the time to freeze, so that he may want to avoid suchactivity. At higher, more energetic levels of activity, the individualmay benefit as a result of an increase in t_(f). These effects on anindividual would take place over the entire range of ambient temperatureand wind velocity. As the ambient temperature increases and the velocitydecreases, the warming effect of q_(b) becomes much greater than thecooling effect of q_(e). This difference reaches its maximum under calmconditions (V=0 mph (0 Km/hr)).

Second, it appears, that because of the offsetting effect of q_(e) andq_(b) at the lower ambient temperatures and the higher velocities wherethe danger of freezing is the greatest, they can be neglected in thedetermination of the time to freeze (t_(f)). This might have beenexpected in light of their similar offsetting effects on wind chilltemperature (T_(wc)) as shown in FIGS. 19A-B and FIGS. 20A-B. At thehigher ambient temperatures and lower velocities, they should beincluded in time to freeze (t_(f)) calculations.

Third, for the low level of activity considered here, the effect of thesolar heat gain (q_(i)) is about 15 times greater than the effect of themetabolic heat gain (q_(b)) at the higher velocity (V=40 mph) and overthe entire temperature range (−40° F.≦T_(a)≦0° F. (−40° C.≦T_(a)≦−17.78°C.)). Thus, sunshine can greatly increase the individual's exposure timebefore facial freezing occurs, but the actual value of t_(f) might bequite small. At the higher ambient temperature (0° F.) and calmconditions (V=0 mph), the beneficial effect of sunshine is so great thatfacial freezing may not occur.

The effects of altitude on the wind chill temperature and time to freezewill now be discussed. Wind chill is primarily the result of a winddriven process that is ambient pressure (P) dependent. One aspect ofwind chill that has escaped attention is its dependency on the ambientpressure and consequently altitude (H). The reason for this may havebeen due to both oversight and the belief that the effect of thispressure was negligible. However, such is not the case. The presentmodel shows that for the same ambient temperature (T_(a)) and windvelocity (V) at all altitudes, increasing altitude will have asignificant effect on the wind chill temperature (T_(wc)) but almost anegligible effect on the time to freeze (t_(f)).

Wind Chill Temperature: The ambient pressure decreases with increasingaltitude and Eq. (13c) shows a corresponding reduction of the forcedconvection coefficient (h_(fc)). Therefore, increasing altitude willresult in a moderation of the wind chill temperature T_(wc). Todemonstrate this effect, calculations of wind chill temperature as afunction of the altitude were made for the basic wind chill components(q_(fc), q_(r)). The evaporative (q_(e)) and metabolic (q_(b))components were not considered because of their negligible andcounteracting effects. Also the solar radiation (q_(i)) was notconsidered, since its absence leads to the more severe wind chilltemperatures which are of greater concern. Calculations of the windchill temperature (T_(wc)) were made using Eq. (18a) with T_(a)=0° F.(−17.78° C.) and −40° F. (−40° C.) and V ranging from 10 to 40 mph(16.09 to 64.37 Km/hr). The range of altitudes (H) considered was fromsea level to the height of Mt. Everest (29,082 ft (8,864.3 m)). Theresults of these calculations, which are plotted in FIG. 25, indicatevery clearly that the wind chill temperature (T_(wc)) increases (ormoderates) with increasing altitude.

FIG. 25A shows the variation of wind chill temperature with altitudewhen T_(a)=0° F. (−17.78° C.). At sea level (H=0 ft (0 m)), for arepresentative coastal city such as Boston, when V=40 mph (64.37 Km/hr),T_(wc)=−86° F. (−65.56° C.). At the same velocity on top of Mt.Washington, T_(wc)=−74.5° F. (−59.17° C.) which is 12° F. (6.67° C.)warmer than for Boston. By comparison, the value on top of Mt. Everestwould be T_(wc)=−34° F. (−36.67° C.) or 52° F. (11.11° C.) warmer. Thislarge difference between Boston and the top of Mt. Everest represents anincrease in T_(wc) of 1.79° F./1000 ft (3.26° C./1000 m) rise inelevation. This is a very significant gradient for what might beconsidered a modest wintertime temperature, but a high wind condition.Consider now Loveland Pass, Colo. at the same wind velocity, V=40 mph.Unlike the current wind chill model, which would predict the same windchill on Loveland Pass as in Boston, this model predicts a T_(wc)=−86°F. in Boston and a T_(wc)=−64° F. (−53.33° C.) for Loveland Pass. This22° F. (12.22° C.) increase in the T_(wc) from the predicted value couldbe of vital importance to downhill skiers in the adjoining Loveland PassSki Area who could readily attain speeds of 40 mph in the absence of anywind.

FIG. 25B, with T_(a)=−40° F. (−40° C.), shows a shift to the left and aconsiderable lowering of the values of T_(wc) from those when T_(a)=0°F. Now when V=40 mph, T_(wc)=−210.39° F. (−134.66° C.) in Boston, a mostunlikely and unbelievable situation, but useful for the sake of thisdiscussion. The corresponding value on top of Mt. Everest would beT_(wc)=−117° F. (−82.78° C.). This translates to an increase in T_(wc)of 3.21° F./1000 ft (5.85° C./1000 m) of elevation rise. Compared to the1.79° F./1000 ft (3.26° C./1000 m) when T_(a)=0° F., this shows that themoderation (warming) of the T_(wc) with altitude (H) becomesincreasingly more important as the ambient temperature decreases.

FIG. 26, which is a plot of this wind chill temperature/altitudegradient over a broad range of temperatures and velocities, provides analternative approach to finding the effect of altitude on T_(wc) ratherthan calculating T_(wc) directly from Eq. (18a). FIG. 26 shows that thegradient increases with decreasing temperature and increasing velocity.From FIG. 26, at realistic conditions of T_(a)=−40° F. (−40° C.) andV=70 mph (112.65 Km/hr) on top of Mt. Washington, the gradient,ΔT_(wc)/1000 ft=3.35° F./1000 ft (6.11° C./1000 m), which means that atthese ambient conditions the wind chill temperature would beΔT_(wc)=3.35° F.×6.2=20.77° F. (11.54° C.) warmer on top of the mountainthan for the same ambient conditions at sea level. The gradients shownin this figure could be used to obtain the value of T_(wc) at any givenaltitude if the corresponding sea level value is known.

FIG. 25 emphasizes the important moderating effect that altitude has onthe wind chill temperature (T_(wc)). This effect on T_(wc) was notconsidered in the prior art wind chill model by Bluestein and Zecher.Perhaps the supposed warming of the T_(wc) experienced by some observersand attributed to their model may have been because the values of T_(wc)they sensed were at an altitude and not at sea level. Possibly somemoderation may be due to Bluestein and Zecher's treatment of a varyingfacial temperature, although the results of FIG. 8 do not show this. Themoderation in wind chill temperature predicted by Bluestein and Zecherappears to be almost entirely due to their incorrect wind reduction. Thefact that this wind reduction was made and the predicted wind chilltemperatures increased as a result of it may have inadvertently resultedin a beneficial effect that partially compensated for their neglect ofany altitude effect.

Time to Freeze: The results in FIG. 25B show the dangerously low valuesof wind chill temperature (T_(wc)) that will be encountered byindividuals when the temperature T_(a)=−40° F. (−40° C.) and thevelocity V≧10 mph (16.09 Km/hr). It is the belief of the inventors thatthese individuals cannot realistically discern the difference between,say, T_(wc)=−80° F. (−62.22° C.) and T_(wc)=−180° F. (−117.78° C.), bothof which are quite possible. This inability to determine the actualT_(wc) combined with its potential to cause a rapid facial freezing ledto the development of Eq. (24) as a means to determine the time tofreeze (t_(f)) in this model. Being able to predict t_(f) for any set ofambient conditions (T_(a), V) not only warns the individual when facialfreezing could occur after exposure, but also frees him from having toknow the actual value of T_(wc). FIG. 25 shows the moderating effect ofincreasing altitude on T_(wc). Similarly, increasing altitude willmoderate or increase the time to freeze (t_(f)). This is shown by usingEq. (22) to make two calculations of the facial temperature decay curve,one at sea level (H=0 ft (0 m)) and one for the maximum altitude(H=29,082 ft (8,864.3 m)), both at the same set of ambient conditions(T_(a), V). For the same reason that the q_(e), q_(b), and q_(i)components were not considered in the effect of altitude on T_(wc), theywere not considered here in the effect of altitude on t_(f).

These temperature decay curves are shown in FIG. 27 for T_(a)=0° F.(−17.78° C.) and V=40 mph (64.37 Km/hr) where t_(f)=2.65 min. at H=0 ft(0 m) and t_(f)=4.55 min. at H=29,082 ft (8,864.3 m), giving a gradientΔt_(f) (4.55−2.65)/29.082=0.065 min/1000 ft (0.214 min/1000 m) ofelevation rise. This means that when T_(a)=0° F. and V=40 mph (64.37Km/hr), the time to freeze will increase by 0.065 min. for every 1000 ftincrease in altitude. For this set of ambient conditions (T_(a)=0° F.,V=40 mph), this seems like a small increase. For example, going from sealevel to a 10,000 ft (3,048.04 m) altitude would increase the time tofreeze by only 0.65 min. or 39 seconds. But because this is not likelyto be the case for all other conditions, calculations of the gradientwere made over the same wide range of ambient temperature (−140°F.≦T_(a)≦25° F. (−95.56° C.≦T_(a)≦−3.89° C.)) and wind velocity (0mph≦V≦160 mph (0 Km/hr≦V≦257.49 Km/hr)) as used in the determination oft_(f) in Eq. (24). Values of the gradient are shown plotted in FIG. 28.The nine curves of the gradient vs. V shown in FIG. 28 were curve fittedusing TableCurve 3D™ to obtain the following expression for the freezingtime gradient

$\begin{matrix}{\frac{\Delta\; t_{f}}{1000} = {a + {b\mspace{11mu}{\ln(V)}} + {c\; T_{a}} + {d\left\lbrack {\ln(V)} \right\rbrack}^{2} + {e\; T_{a}^{2}} + {f\; T_{a}{\ln(V)}} + {g\left\lbrack {\ln(V)} \right\rbrack}^{3} + {h\; T_{a}^{3}} + {i\; T_{a}^{2}{\ln(V)}} + {j\;{T_{a}\left\lbrack {\ln(V)} \right\rbrack}^{2}}}} & (32)\end{matrix}$where a=0.17046053, b=−0.037699401, c=0.0011765359, d=0.0022511491,e=6.1594515×10⁻⁶, f=−9.6552578×10⁻⁵, g=3.3382238×10⁻⁶, h=1.7864164×10⁻⁸,i=1.2133567×10⁻⁹, j=−1.4109763×10⁻⁷ and where the correlationcoefficient is r²=0.99566554. Each of the nine curves in FIG. 28 shows acontinual and rapid rise in the value of the gradient if the velocitywere to decrease below V=1 mph (1.61 Km/hr) and approach V=0 mph (0Km/hr). This is a reflection of what has already been shown in FIG. 14,where the time to freeze (t_(f)) is very sensitive to any increase invelocity from the calm condition (V=0 mph). FIG. 28 shows that thelarger, more beneficial values of the gradient occur at the warmertemperatures and the lower velocities, which are conditions where facialfreezing is of lesser concern. Unfortunately, at the lower temperaturesand higher velocities where freezing is much more likely, thesegradients are much smaller. Since the actual value of the time incrementΔt_(f) depends not only on the ambient conditions (T_(a), V), but alsoupon the altitude (H), the larger values of Δt_(f) will occur at thehigher altitudes. One such place is Loveland Pass, Colo., which, at analtitude of 11,990 ft (3,654.6 m), represents one of the higheraltitudes in the lower 48 states. For possible ambient conditions ofT_(a)=0° F. and V=10 mph (16.09 Km/hr) there, the gradient Δt_(f)/1000ft=0.096 min (Δt_(f)/1000 m=0.31 min) from Eq. (32) or directly fromFIG. 28. Thus, the change in time to freeze due to the altitude isΔt_(f)=0.096×11.99=1.15 min. This would be close to the maximum value ofΔt_(f) that would be encountered across the 48 states. For all lowertemperatures (T_(a)), higher velocities (V) and lower altitudes (H),Δt_(f) would be less than 1.15 min. If a value of Δt_(f) is much lessthan this, then an altitude correction of t_(f) may not be necessary.However, if exactness is required, the value of Δt_(f) should becalculated from Eq. (32) and then added to the sea level value of t_(f)from Eq. (24).

The following is a description and demonstration of a method forcalculating the wind chill temperature (T_(wc)), the time varying facialtemperature (T_(f)) and the facial time to freeze (t_(f)) for anindividual exposed to wintertime ambient conditions (T_(a), V) accordingto the methods of the present invention. The following information mustbe known or specified before proceeding with the calculation of thosethree quantities: (1) Ambient temperature (T_(a), ° F.), (2) Windvelocity (V, mph) at the NWS 10 m level, (3) Latitude (LAT) atindividual's location, (4) Location altitude (H, ft), (5) Presence (G>0)or absence (G=0) of sunshine, (with sunshine, G is determined from Table2 for the known latitude region (LAT)), (6) Metabolic heat generation(M_(act)) determined from Table 1 for an individual's known physicalactivity, with M_(act), the evaporation flux rate ({dot over (w)}) canbe determined, (7) Distance (D) of an individual from the nearest windobstruction; with D, the wind reduction factor (WRF) can be determined(see below). The first five of these items are automatically known oncethe weather conditions and location are specified. The remaining tworelate entirely to the individual being examined.

In demonstrating the method used to calculate T_(wc), T_(f) and t_(f),consider the extreme weather conditions that existed on top of Mt.Washington, N.H. on Jan. 22, 2003, when the ambient temperature dippedto −34° F. (−36.67° C.) and the wind speed reached 142 mph (228.52Km/hr). Mt. Washington has an altitude of 6,200 ft (1,889.78 m) and liesat 440.14 north latitude. At this excessive wind speed, stormyconditions almost certainly prevailed. If these stormy conditionsoccurred during daytime hours, then sunshine may have been at a maximum(G>0) or a minimum (G≅0). With this description, the followinginformation is known: T_(a)=−34° F.; H=6,200 ft; V=142 mph; G≅0 or 42.66Btu/hr-ft² (0 or 134.58 W/m²) from Table 2 using LAT; LAT=44°.14′.

The calculations were made for an individual presumed to be a member ofthe weather station located on top of the mountain. The individual wasassumed to be fully clothed, standing outside, fully exposed to thewind, yet removed from any turbulence effect produced by the stationitself. Under more normal conditions, the individual's activity might bedescribed as “standing relaxed”, using the terminology of Table 1.However, an individual standing in a 142 mph wind is not likely to berelaxed. On the contrary, he will be expending a great deal of energyjust to remain upright. This suggests that the metabolic heat generationmay be more comparable to that of a downhill skier. Thus from Table 1,M_(act)=96 Btu/hr-ft² (302.84 W/m²) and since M_(act)≧40 Btu/hr-ft²(126.18 W/m²), {dot over (w)}=0.02183 lbm/hr-ft² (0.1066 kg/hr-m²).Finally, the leeward distance (D) does not exist, since the individualis not downstream of the station. Rather D must be replaced by thewind/surface contact distance (l) which, near the mountain top, would beexpected to be very small. Suppose l=100 ft (30.48 m), but because thesurface is not likely to be smooth, l was assumed to be 200 ft (60.96 m)to account for surface roughness and boundary layer buildup. Therefore,D=l=200 ft. With l=200 ft and V=142 mph, the boundary layer edge (δ)from Eq. (33), below, is δ=(200)(0.0589)[(142)(200)]^(−0.2)=1.52 ft(0.46 m). Since δ<y (head height) and where y=5 ft (1.52 m), then fromEq. (35b), below, WRF=0. With the above values of T_(a), V, H, G,M_(act), {dot over (w)} and WRF known, the values of T_(wc), T_(f) vs.time and t_(f) can be determined.

Calculation of the Wind Chill Temperature (T_(wc)): Calculation ofT_(wc) from Eq. (18a) must be preceded by a calculation of the exponentφ from Eq. (18d). This may be accomplished for the individual in minimumsunshine (G≅0) and in maximum sunshine (G=42.66 Btu/hr-ft²).

-   -   G≅0        -   From Eq. (18d), φ=0.506257        -   From Eq. (18a), T_(wc)=−296° F. (−182.22° C.)    -   G=42.66 Btu/hr-ft² (134.58 W/m²)        -   From Eq. (18d), φ=0.506257        -   From Eq. (18a), T_(wc)=−294° F. (−181.11° C.)

These very cold temperatures are considered to be the correct values ofT_(wc) on top of Mt. Washington. They are values that lie well beyondthe range of applicability of the currently used wind chill model which,among other things, cannot account for the effects of altitude orsunshine. This can be clearly demonstrated in the following manner. FIG.8 shows that the Bluestein and Zecher results, without their incorrectwind reduction, yields results that are essentially those of Siple andPassel. This means that Eq. (27a), which is the analytical expression ofthe Siple and Passel results, should provide the prior art model'sprediction of the T_(wc) on top of Mt. Washington. With T_(a)=−34° F.and V=142 mph, Eq. (27a) predicts this temperature as T_(wc)=−60.4° F.(−51.33° C.), an unbelievable underprediction of the actual value,T_(wc)=−296° F. The small 2° F. (1.11° C.) increase in T_(wc), whenmaximum sunshine is present, agrees with the findings of FIG. 22A-Bwhere the presence of sunshine at a high velocity (V) and a lowtemperature (T_(a)) resulted in a very small increase in T_(wc). Thissmall increase in T_(wc) due to sunshine is obviously insignificant inthese extreme ambient conditions. The effect of altitude, which has beenneglected in all wind chill predictions of the prior art, does have asignificant effect. The error in T_(wc) attributed to neglecting thiseffect of altitude can be determined from the wind chill temperaturegradient of FIG. 26. With T_(a)=−34° F. and V=142 mph, the wind chilltemperature gradient is ΔT_(wc)/1000 ft=4.09° F. (7.45° C./1000 m). Thismeans that the above correct value of T_(wc)=−294° F. in the presence offull sunshine, would have been 25.36° F. (4.09×6.2) colder atT_(wc)=−319.36° F. (−195.20° C.) if the effect of Mt. Washington'saltitude had not been considered.

Calculation of the Time to Freeze: Eq. (24) is the expedient way ofcalculating the time to freeze (t_(f)). Results from using Eq. (22) forthe facial temperature (T_(f)) were the basis for developing thisexpression for the time to freeze. Although a lengthier approach, Eq.(22) will be used first to calculate the time dependent decrease infacial temperature (T_(f)) from its initial value of 91.4° F. (33° C.)to the time of facial freezing (32° F. (0° C.)). Calculation of t_(f)from Eq. (22) starts with the initial facial temperature T_(f)^(m)=91.4° F. at time zero (m=0) and then calculates the facialtemperature, T_(f) ^(m+Δt), at a later time increment (Δt). Using whathas been found to be an optimum time increment Δt=1 second, T_(f) iscalculated after one second. Repeated calculations of T_(f), where thevalue at the end of the time increment becomes the starting value on thenext, continues until T_(f) reaches 32° F. Results of these calculationsare shown in FIG. 29 in the absence (G=0 Btu/hr-ft² (0 W/m²)) andpresence (G=42.66 Btu/hr-ft² (134.58 W/m²)) of sunshine. The interceptpoints of these facial temperature decay curves with the freezing(T_(f)=32° F. (0° C.)) line, shows that t_(f)=1.0 min. when sunshine isabsent and that t_(f)=1.035 min. when it is present. A comparison ofthese values shows that sunshine at these extreme ambient conditionsdoes little (0.035 min. or 2.1 seconds) in extending the time to freeze.This might have been expected, since this effect of sunshine has beenshown to be minimal on T_(wc) at very extreme conditions as shown inFIGS. 22A-B.

Calculations of t_(f) for Mt. Washington using Eq. (24) give t_(f)=0.897min. with no sunshine (G=0) and t_(f)=0.9155 min. with sunshine (G>0).Since Eq. (24) is restricted to sea level conditions, these sea levelbased values of t_(f) must be corrected for the Mt. Washington altitude.From Eq. (32) or FIG. 28 when T_(a)=−34° F. and V=142 mph, the time tofreeze gradient is Δt_(f)/1000 ft=0.022 min. (Δt_(f)/1000 m=0.07 min.).Therefore, the increase in time to freeze on top of Mt. Washingtoncompared with that at sea level is 0.14 min. (0.022 min.×6.2). Addingthis time increment to the above values gives the altitude-correctedvalues of Eq. (24) as t_(f)=1.037 min. with no sunshine and thatt_(f)=1.056 min. with sunshine. These more readily determined values oft_(f) are only 3.7% and 1.98% larger than the corresponding actualvalues (1.0 min., 1.035 min.) from Eq. (22). Therefore Eq. (24) for sealevel conditions along with an altitude correction from either Eq. (32)or FIG. 28 provides a quick and relatively accurate value of t_(f).

The following observations are made regarding the wind chill modeldisclosed herein. First, the Siple and Passel experiment was not greatlyflawed as has been thought by several investigators. A closerexamination of this experiment reveals that the only error of anyconsequence was their assumption of a constant skin temperature of 33°C. (91.4° F.) during the entire exposure time. In reality, as theexposure time increases, the facial skin temperature will decrease andthe wind chill temperature sensed by the individual will increase orbecome warmer. This was correctly recognized by the critics. However,their argument against the assumption of a constant 33° C. (91.4° C.)temperature seems to be that, over an extended exposure time, theindividual was somehow being deprived of feeling warmer. Actually, thisargument is irrelevant, because the individual is much more likely to beinterested in the wind chill temperature at the initial moment ofexposure, when the face temperature is realistically near 91.4° F. (33°C.), than at times later, when his facial temperature may be plummetingand he becomes preoccupied with thoughts on how to avoid facialfreezing.

Second, Siple and Passel's assumption of a constant skin temperature wasa valid criticism. Bluestein and Zecher's development of their prior artwind chill model was an attempt to show a moderation (warming) of theSiple and Passel wind chill temperatures by allowing the skintemperature to vary rather than to remain constant. But, the Bluesteinand Zecher model shows at most a 2° F. (1.11° C.) moderation as a resultof the skin temperature variation. At very low temperatures and highvelocities, the Bluestein and Zecher values show no moderation. Ratherthey show a −1° F. cooling relative to the Siple and Passel values. Thebalance of their total 15° F. (8.33° C.) increase in the wind chilltemperature was the result of the head level wind reduction based ontheir erroneous assumption that the NWS 10 m velocity value is always50% greater than that at head level. The conclusion reached herein isthat without their incorrect wind reduction and with a seeminglyincorrect skin temperature correction, the Bluestein and Zecher valuesof the wind chill temperature are really no different from the Siple andPassel values they were intended to correct.

Another observation is that the wind chill model developed here iscomplete, accurate and more adaptable than the prior art Bluestein andZecher model in use as of this writing. Like all previous attempts atwind chill prediction, it is based upon the two basic skin heat lossprocesses, namely radiation (q_(r)), which is always present, and forcedconvection (q_(fc)), which is present only under windy conditions. Thesetwo heat losses are the only ones that have been considered in previouswind chill analyses. What makes the wind chill model of the presentinvention complete is that it also incorporates both the beneficial(warming) effect of an individual's physical activity and that ofsunshine when it is present, both of which are considered as heat gainsat the facial surface. Physical activity produces a metabolic heat flow(q_(b)) from the body core and a solar radiation heat flow (q_(i)) fromsunshine. The physical activity produces a third facial surface heatloss due to sweat evaporation (q_(e)). The model is unique in that itincludes (a) the very important effect of altitude (H), a variable whichuntil now has not been considered, and (b) the capability to determinein each case a correct wind reduction factor (WRF), rather thanuniversally applying an incorrect one. The greater accuracy of thismodel stems not only from the addition of the above terms (q_(b),q_(i)), but also from the means by which they were determined. Themetabolic heat (q_(b)) from the body core to the facial surface wasbased upon an accurately derived thermal conductance term from a humanthermoregulation model by Havenith. The solar radiation heat (q_(i)) wasbased upon monthly insolation data for 221 cities across the 50 statesas compiled by the U.S. Department of Energy. The WRF can beanalytically determined for each case knowing the individual's locationrelative to an upwind obstruction, see discussion below. Finally, themodel is adaptable over the widest range of ambient temperatures (−140°F.≦T_(a)≦25° F. (−95.56° C.≦T_(a)≦−3.89° C.)), wind velocities (0mph≦V≦160 mph (0 Km/hr≦V≦257.49 Km/hr)) and altitudes (0 ft<H<29,082 ft(0 m<H<8,864.3 m)) worldwide.

Another observation is that verification of this model consisted ofcomparing this model's predicted values of wind chill temperature(T_(wc)), facial temperature (T_(f)) and time to freeze (t_(f)) withexisting experimental data. First, the predicted values of the windchill temperature from Eq. (18a) were compared with those of Siple andPassel. This also becomes a comparison with the Bluestein and Zechervalues when their incorrect wind reduction is removed. The comparisonwas made for the following conditions that existed at their location inAntarctica at the time of their experiment: sea level (H=0 ft (0 m))altitude, no wind reduction (WRF=0) at the height of their containerabove ground level, no evaporative heat loss (q_(e)=0) and no metabolicheat gain (q_(b)=0), since neither of these applied, and no solarradiation (q_(i)=0) since the experiment was conducted in darkness.Comparison of this model's predicted values of T_(wc) with those forSiple and Passel values in FIG. 7B shows that the predicted values aremuch colder, particularly at the higher wind velocities. This is incontrast to the belief of many investigators that the Siple and Passeltemperatures were already colder than the actual values because of theirconstant temperature assumption. Additionally, the predicted values ofthe facial temperature from Eq. (22) were found to be in very goodagreement with facial temperature test data obtained by Buettner,LeBlanc et al., the home freezer experiments performed by the inventors,and to some extent with the Discovery Channel™ Experiment and theresults of Adamenko and Khairullin. This agreement gives credence to theexpression for wind chill temperature in Eq. (18a) since it and Eq. (22)for facial temperature are based on the same five heat transferprocesses. Finally, this agreement gives confidence in predictions ofthe time to freeze from Eq. (24), since this equation was derived fromresults using Eq. (22). No known data were available to make acomparison with predicted values of the time to freeze. However, theactions taken by some concerned contestants in the 2003/2004 WorldDownhill Ski Championships clearly indicated the capability of Eq. (24)to correctly predict the time to freeze in a real life situation.

Still another observation is that the wind reduction at head leveldepends upon the extent to which an individual's head is immersed ineither a wind-generated boundary layer along the surface, upon which theindividual is positioned, or within a wind-generated turbulence regionon the leeward side of an obstruction upwind of the individual. Ineither case, the distance (D) between the individual and the obstructionmust be known. It is this distance and the known wind velocity (V) thatultimately determines the value of the wind reduction factor (WRF).Considering the entire populace in all weather conditions, there are aninfinite number of D and V combinations. Consequently, an infinitenumber of WRFs are possible. It is for this reason that theacross-the-board application of a WRF=0.33 in the prior art wind chillmodel is in error. This detailed description further describes aprocedure, below, for determining a more exact WRF in terms of D and V.Applying this procedure, average values of WRF were determined for bothan urban and a rural area over the 20 mph≦V≦70 mph (32.19 Km/hr≦V≦112.65Km/hr) range. For an urban area, WRF=0 for all velocities. For the ruralareas the average values of WRF are 0.055 for V=20 mph and 0.072 forV=70 mph. The fact that these two areas may encompass more than 95% ofthe populace, and since these values of WRF are small and nearlynon-existent, emphasizes the error introduced in the current model whena WRF=0.33 is being used.

Yet another observation is that physical activity produces a metabolicheat flow (q_(b)) to the skin surface and evaporative heat loss (q_(e))from the surface. Their combined effect on the wind chill temperature(T_(wc)) and the time to freeze (t_(f)) is small and nearly offsettingat low temperatures (T_(a)) and high wind velocities (V), when facialfreezing is of the greatest concern. When freezing is of lesser concernat high temperatures and low velocities, q_(b) does provide a beneficialincrease in T_(wc) and t_(f) above the opposing effect of q_(e).Physical activity will have a completely different effect on the clothedportion of the body. The clothing will act as a barrier to prevent orslow the metabolic heat loss from the body. It will also act as anabsorbent of the perspiration preventing its evaporation and itsadditional heat flow from the body. The additive effect of these twoheat losses may explain why physical activity is known to have a warmingeffect on the clothed body. This warming effect plays no part in thewind chill temperature as sensed by the facial surface, nor in thepossibility of facial freezing.

Another observation is that sunshine has a much greater moderatingeffect on wind chill temperature (T_(wc)) and time to freeze (t_(f))than physical activity. This greater moderation by sunshine extends overa relatively wide range of ambient conditions. Unfortunately, thebenefit of this moderation is the least at low ambient temperatures andhigh velocities, where facial freezing is a real possibility. However,under calm (V=0 mph) conditions and relatively high temperatures such asT_(a)=0° F., the effect of sunshine is so great that facial freezing maynever occur. The conclusion is that the presence of sunshine results inits greatest beneficial effects at the lower velocities and the highertemperatures.

Furthermore, increasing altitude has the effect of increasing the windchill temperature (T_(wc)) and the time to freeze (t_(f)). Calculationsusing this model gave rise to an incremental increase in the T_(wc) per1000 ft increase in altitude. This gradient increases rapidly withdecreasing ambient temperature (T_(a)) and increasing velocity (V). Whatthis means is that the moderating or beneficial effect of altitude onT_(wc) is the greatest at the lower temperatures and higher velocitieswhere it is needed the most. Even though altitude has this great effecton T_(wc), its effect on the time to freeze (t_(f)) is just the oppositeand quite small. Calculations of a corresponding time to freezegradient, Δt_(f) per 1000 ft, show that it decreases as the velocity (V)increases and as the ambient temperature (T_(a)) decreases.Unfortunately, the benefit of this altitude increase is least atvelocities and temperatures where it is most needed. Furthermore, theincrease in t_(f) due to altitude is very small. At even lowertemperatures (T_(a)) and higher velocities (V), this incrementalincrease in t_(f) with altitude would be even smaller, and as such canpossibly be neglected.

Procedure for Determining Wind Chill Factor: In wind chill calculations,the ambient temperature (T_(a)) is presumed to be known, but determiningthe actual wind velocity at head level may be difficult if notimpossible except in special situations. This is because the actual windvelocity depends upon whether or not the individual's head is immersedin a wind-generated turbulence. If it is not, the head is exposed to thefree-stream velocity (V), which is assumed to be that at the NWS 10 mlevel. The wind reduction factor (WRF) is defined as the differencebetween the free-stream velocity and the head level velocity divided bythe free-stream velocity. Thus, there would be no wind reduction whenthe free-stream velocity equals the head level velocity, i.e., WRF=0.

The problem here is that the 10 m level, except in specific instances,does not represent the correct height upon which to determine the WRF.Therefore the current practice of assuming the free-stream velocity asbeing 50% greater than that at head level in all calculations of thewind chill temperature is incorrect. The following discussion will showwhy the WRF must be determined from the depth of the velocity boundarylayer and not from the height of the NWS 10 m velocity sensor, unlessthe latter happens to be exactly at the boundary layer edge. Windreduction becomes necessary when the head is immersed in either: (a) aturbulent region on the leeward side of a wind obstruction, or in (b) aturbulent boundary layer generated by the wind. The latter, which ispossibly the more likely situation to occur, fortunately is the one thatlends itself more easily to analysis, providing the followinginformation is known: (1) the location of the boundary layer edgerelative to the individual's head, and (2) the velocity profile withinthe boundary layer.

Unfortunately, this information is so dependant upon an individual'ssurroundings that an evaluation of the WRF may not always be possible.But this is no reason for universally applying an incorrect value of WRFto all individuals in the listening area. Determination of this boundarylayer is explained in the following paragraphs.

Wind blowing along a surface experiences a retarding action by frictionover a layer called the velocity boundary layer. Within this layer, thevelocity increases from zero at the surface to the free-stream value (V)at the boundary layer edge (δ). If an individual's head is within theboundary layer, it will experience a velocity (v) less than thefree-stream value (V) and consequently a WRF>0. The thickness (δ) ofthis boundary layer is a function of the free-stream velocity, the air'skinematic viscosity (ν) and most importantly the length (l) that thewind is in contact with the surface. Another variable affecting thethickness is the surface roughness, but this is not easily determined.What is known, is that this roughness guarantees that the flow in thelayer will be turbulent and that an increase in roughness will increaseδ. Based on all this, it can be stated that an individual exposed to afree-stream velocity (V) will encounter a turbulent boundary layerthickness (δ) that is dependant upon the individual's surroundings, suchas the surface roughness on his windward side and upon the wind/surfacecontact length (l).

FIG. 30 is a diagram illustrating a wind-generated turbulent boundarylayer according to the present invention. Referring to FIG. 30, consideran obstruction such as a fence, a tree or a group of buildings locatedat a distance (D) from an individual facing into the oncoming wind. Thewind at velocity V approaching the obstruction will separate from thesurface and flow over the obstruction to produce a vortex type ofseparation region on its downstream or leeward side. This separated flowregion will reattach to the surface at a distance (x) from theobstruction. It is the distance l=D−x which is the critical length inthe determination of the boundary layer thickness (δ) at theindividual's location and whether or not the individual's head isimmersed in it. If it is not, then the head height, which is thedistance of the head base above the surface, is greater than δ and thehead is exposed to free-stream conditions so that WRF=0. This is thesituation illustrated in FIG. 30. If the head is immersed, the headheight is less than δ. In this case, it is δ and not the standard 10 mheight of the NWS 10 m velocity sensor that must be used in thecalculation of the WRF, unless of course, δ actually is 10 m. Thus, aheight of 10 m represents a very special case. This illustrates theimportance of knowing the value of δ for a range of velocities (V) andwind/surface contact distances (l) that might be encountered.

Schlichting, H., “Boundary Layer Theory”, McGraw-Hill, p. 42, 1979,discloses that the variation in the boundary layer thickness (δ) for theturbulent flow is:

$\begin{matrix}\begin{matrix}{\frac{\delta}{l} = {{.37}\left( \frac{V\; l}{v} \right)^{{- 1}/5}}} \\{= {{.37}\left( {Re}_{l} \right)^{{- 1}/5}}}\end{matrix} & (33)\end{matrix}$where the kinematic viscosity (ν) of air is 150×10⁻⁶ ft²/s (13.94×10⁻⁶m²/s). With the velocity in the Reynolds number (Re_(l)) expressed inmph, Eq. (33) becomes:

$\begin{matrix}{\frac{\delta}{l} = {{.0589}\left( {V\; l} \right)^{- 0.2}}} & (34)\end{matrix}$

FIG. 31 shows 5 as a function of V and l as expressed by Eq. (34).Assume the base of an individual's head is at the 5 ft (1.52 m) level.Each circled intersect point represents the maximum wind/surface contactdistance (l_(max)) at a given velocity where the base of theindividual's head would be at the boundary layer edge. For example, thismeans that when V=40 mph (64.37 km/hr), the maximum distance (l_(max))is 648 ft (197.51 m) and the head is above the boundary layer edge, andthe WRF=0. For distances greater than (l_(max)), the individual's headis partially or completely immersed in the boundary layer, and althoughWRF>0, its actual value may be very small. As V increases, l_(max)increases. This is more clearly demonstrated in FIG. 32, where thel_(max) curve is a cross plot of the intersect points in FIG. 31. At anygiven velocity in FIG. 32, for values of l<l_(max), there is noimmersion of the head and WRF=0; for l>l_(max) there is immersion, andWRF>0. It should be pointed out that Eq. (34) applies to a smoothsurface such as a paved road, sidewalk, or an airport runway. For othersurfaces where a roughness exists due to small objects or vegetation,values of δ would be expected to be slightly larger. This small increasewould shift the curves of FIG. 31 slightly upward, thus reducing thel_(max) value at a given velocity. This small increase would also shiftthe (l_(max)) curve of FIG. 32 slightly downward. Lacking theinformation required to correct for this slight difference, the l_(max)curve of FIG. 32 is presumed to be sufficiently accurate for allsurfaces.

FIG. 32 shows l_(max) as the defining distance downstream of anobstruction that determines whether or not an individual's head isimmersed. Because it is advantageous to reference this defining distanceto the individual's actual distance (D) from the obstruction, D isdetermined by adding the separated flow reattachment distance (x) tol_(max) as shown in FIG. 30. The problem here is that x is not a fixedquantity, but rather increases with the height of the obstruction, andincreases as V increases. At this point, the assumption was made that anobstruction such as a tree or building will produce a downstream flowseparation distance (x) of 100 ft and 30 ft (30.48 m and 9.14 m) atvelocities of 70 mph (112.65 Km/hr) and 20 mph (32.19 Km/hr),respectively. Linearly spreading these distances over the velocity rangeand adding them to the l_(max) distances in FIG. 32 gives curve D, theapproximate distance of the individual from the obstruction. Thisdistance D is approximate because the two separation distances (100 ft,30 ft) for the two velocities (70 mph, 20 mph) apply to one specificobstruction height and not for all heights as assumed here. However, thevariation in distance x with obstruction height is believed to be asmall fraction of l_(max), so that ignoring this effect should notresult in any significant error. Distance D can now be considered forall conditions as an approximate, although realistic, distance betweenan individual and the obstruction that determines the flow field in hispresence. Curve D, now replacing curve l_(max), separates the regionWRF>0 above it from the WRF=0 region below. From this, one concludesthat if the individual is within distance D of 600 ft (182.88 m) to 850ft (259.08 m) from an obstruction over the 20 mph to 70 mph velocityrange, he will still be exposed to free-stream conditions, that isWRF=0. It is believed that this represents a large majority of real lifesituations in which the currently used 50% wind reduction is incorrectlyapplied.

With reference to FIG. 32, if an individual at a given velocity (V) isat a distance greater than D from an obstruction, his head will bepartially or completely within the boundary layer. In this case, the WRFmust be determined. The WRF can then be computed from the velocityprofile within the boundary layer. From Schlichting, this profile in aturbulent boundary layer is,

$\begin{matrix}{\frac{v}{V} = \left( \frac{y}{\delta} \right)^{1/n}} & (35)\end{matrix}$where, in this case, v is the velocity at head level, V is thefree-stream velocity at the boundary layer edge (δ), y is the headheight above ground level, and where the exponent 1/n depends upon thesurface roughness and the free-stream velocity. From its definition andusing Eq. (35), the WRF can be determined in the following manner:

$\begin{matrix}{{{{{if}\mspace{14mu}\delta} > y},\begin{matrix}{{WRF} = \frac{V - v}{V}} \\{= {1 - \left( \frac{y}{\delta} \right)^{1/n}}}\end{matrix}}{{and},}} & \left( {36a} \right) \\{{{{if}\mspace{14mu}\delta} \leq y},{{WRF} = 0}} & \left( {36b} \right)\end{matrix}$Calculating an accurate value of this WRF is complicated by theexponent's dependency upon the surface condition and the magnitude ofthe wind speed. Consider first the surface condition. Schlichting andalso, Eshbach, O. W., Handbook of engineering fundamentals, John Wiley &Sons, Inc., p. 1-142 and 7-119, 1952, state that for smooth surfaces ofthe type encountered in wind chill calculations such as a sidewalk,street, airport runway, or even a frozen lake, the exponent would be1/7, borrowing from the “ 1/7 power law” for pipe flow. This value issmaller than the corresponding value (1/4.76) used by Steadman based onmeasurements in the Saskatoon area. Based on Steadman's values, the WRFat head height (y=5 ft (1.52 m)) is

$\begin{matrix}\begin{matrix}{{WRF} = {1 - \left( \frac{5}{33} \right)^{1/4.76}}} \\{= 0.33}\end{matrix} & (37)\end{matrix}$This WRF of 0.33 corresponds to the universal 50% reduction in the NWS10 m wind speed used in the prior art Bluestein and Zecher wind chillmodel.

If the 33 ft value for δ as used in Eq. (37) was actually at the edge ofthe boundary layer in the flow field during the Steadman measurements,then his larger exponent (1/4.76) could have been expected if themeasurements were made over a prairie surface exhibiting a certainamount of roughness. If that was the case, then Eq. (37) does provide acorrect WRF for an individual on a similar surface, but only if theindividual is at a sufficient distance D from an obstruction such thatthe boundary layer thickness (δ) at the individual's location is exactly33 ft (10.1 m). This is the only instance in which the current WRF of0.33 can be correctly applied. In all other cases, this correction istoo severe. This can be demonstrated as follows: For velocities (V)spanning the wind chill range, select values of l and calculate (fromEq. (34). Then calculate WRF from Eq. (36a) using y=5 ft (1.52 m) andSteadman's exponent. The results are plotted in FIG. 33 where thewind/surface contact distance (l) has been replaced by distance D froman obstruction. They show that at V=25 mph (40.23 Km/hr) the individualmust be at a distance D>6,095 ft (1,857.78 m) from an obstruction if theWRF is to be 0.33. For V=70 mph, the corresponding distance would be7,884 ft (2,403.07 m). These very large distances could exist insparsely populated rural areas, but are not too likely in urban areas.The message gathered from FIG. 33 is that for the lower distancesassociated with urban areas, the WRF is much less than 0.33. It hasalready been shown that individuals within distance of 600 ft (182.88 m)to 850 ft (259.08 m) of an obstruction will not experience a windreduction; for them, WRF=0. All other individuals in the listening areaat greater distances will experience some reduction, though it will besmall. It would seem reasonable to assume that more than 95% of thelistening public may be within a 1,500 ft (457.21 m) distance(approximately ¼ mile) from an obstruction. From FIG. 33, individuals atD=1,500 ft (457.21 m) will experience wind reduction factors varyingfrom 0.11 at V=70 mph to 0.144 at V=25 mph. The average values of WRFfor the above 95% of the listening public between 600 ft (182.88 m) and1,500 ft (457.21 m) would be 0.055 at V=70 mph and 0.072 at V=25 mph.These values are so much lower than the currently used value of 0.33that there is an obvious need to use these lower and more exact values.Continuing to use the current value (0.33) only continues to mislead thelistening public, because the reported wind chill temperature will bemuch warmer than its actual value. To the individual, this implies alesser concern for facial freezing, when in reality a greater dangerexists. Because of the near impossibility of determining a realisticaverage value of WRF for all individuals at all locations in thelistening area, it might be better to disregard any wind reduction.Neglecting it could be viewed as a safety feature for the listeningpublic, since it would effectively predict facial freezing before itactually occurs. Thus, assuming WRF=0 for all conditions would be asimple and reasonable solution.

The above discussion and the results shown in FIG. 33 were based on theassumption that the 33 ft (10.1 m) height in Steadman's equation wasactually at the boundary layer edge. It is not known at what wind speed(V) the measurements were made in the development of his velocityprofile. Suppose V was 25 mph. From FIG. 33, this means that distance(D) upstream of the instrumentation would have to have been about 6,095ft. It is important to note that the distance was about 6,095 ft,because of the uncertainty as to how well Eq. (33) represents thesurface conditions where the measurements were made, since this equationapplies strictly to a smooth surface. What it does mean is that if V was25 mph and the open air distance upstream of the instrumentation wasknown to be approximately this distance, then the boundary layer edgewas probably very close to the 33 ft height. This same reasoning would,of course, apply if the measurements had been conducted at a velocityother than V=25 mph (40.23 Km/hr) and a correspondingly differentdistance.

One way of determining whether or not the 33 ft height represents theboundary layer edge (δ) would be the measurement of the height aboveground level where the velocity ceases to increase with height; thisheight would then be the edge. This approach may have been taken duringthe Saskatoon measurements to insure that the 33 ft height in Eq. (37)represents the true boundary layer edge. If it does not, then Steadman'svelocity profile expression is incorrect and its usage would result inan incorrect value for the WRF. This may no longer be of much importanceif the use of a WRF is discontinued, as is suggested in the presentmodel.

There may be certain situations where there is a need to determine aWRF. If Steadman's exponent was incorrectly determined, an alternativewould be to use the 1/7 exponent. Although this exponent supposedlyapplies to smooth surfaces such as an airport runway, a blacktop road, ahard packed sand beach or a snow-free frozen lake surface, it shouldprovide a good approximation of the WRF even for surfaces with minorroughness. However, even the 1/7 exponent has its limitation. Eshbachnotes that it is applicable only up to a Reynolds number (Re_(l)) of3×10⁷, above which the exponent will progressively decrease to ⅛, 1/9etc., as the Reynolds number increases. Based on the definition of theReynolds number in Eq. (33), this means that the 1/7 exponent is validup to a wind/surface contact length (l) of 153 ft (46.63 m) at avelocity (V) of 20 mph and only 44 ft (13.41 m) at a velocity of 70 mph.These are very short distances and one would expect numerous cases wherethe wind/surface contact length far exceeds these values. If so, theReynolds number would be much greater than 3×10⁷ and the exponent lessthan 1/7. The exact value of the exponent is unknown. What is known isthat the WRF decreases as the exponent decreases. Suppose the boundarylayer edge (δ) is 5 ft above an individual's head at y=5 ft. Then(y/δ)=0.5. With this value in Eq. (36a), the WRF is calculated and shownin FIG. 34 as a function of the exponent's denominator (n). The resultsshow that for a smooth surface when n is 7 or higher, the WRF is muchless than the 0.136 value obtained using Steadman's exponent. In fact ifn=10, the WRF is one-half of this value. What was considered a goodreason for neglecting wind reduction before, based on the FIG. 33results using Steadman's exponent, is an even better reason now, whenconsidering the likelihood that the exponent denominator (n) will begreater than 7 for the larger Reynolds numbers that will be encountered.One concludes from all this that, irrespective of the type of surface,any wind speed reduction, if it exists, is negligible. Consequently,unless there is a specific reason to consider wind reduction, it shouldbe neglected by assuming WRF=0. As already stated, any error due to thisneglect is on the side of safety, as far as the listening public isconcerned.

Another aspect pertaining to the computation of a WRF is the location ofthe NWS 10 m sensor. It is presumed to measure the true free-streamvelocity. To do so, it must be located at or above the boundary layeredge that exists at its airport location for all wind conditions. Sims,C., Personal communication, National Weather Service, Duluth, Minn.,2001, provided the following specific information on the NWS 10 m sensorwhich is part of the ASOS/AWOS (Automated Surface ObservingSystem/Automated Weather Observing System) site: “It will be mounted 30to 33 feet (9.14 to 10.1 m) above the average ground height within aradius of 500 ft (152.4 m)”. In addition, Sims states: “The sensorheight shall not exceed 33 ft except as necessary to: (a) be at least 15ft (4.57 m) above the height of any obstruction (e.g., vegetation,buildings, etc.) within a 500 ft (152.4 m) radius, and (b) if practicalbe at least 10 ft (3.1 m) higher than the height of any obstructionoutside the 500 ft radius, but within a 1000 ft (304.8 m) radius of thewind sensor. An object is considered to be an obstruction if theincluded lateral angle from the sensor to the ends of the object is 10degrees or more.”

First, consider the case where there is no obstruction within 500 ft ofthe sensor. To determine if the sensor is located above the boundarylayer edge at the sensor location, the layer's thickness (δ) wascalculated from Eq. (34) using a wind/surface contact length (l) equalto the 500 ft radius. With a wind velocity V=20 mph and 70 mph directedtowards the sensor, the corresponding boundary layer thickness would beδ=4.7 ft (1.43 m) and 3.6 ft (1.1 m), respectively. With the sensor at30 to 33 ft above the ground height, it would be well within thefree-stream region. Second, in the situation where there is anobstruction within the 500 ft radius, the concern no longer centers onthe wind/surface boundary layer thickness, but rather on the size of theturbulent flow region downstream of the obstruction. The NWS 10 m sensormust be above this region to experience free-stream conditions. Becausethe vertical dimension of the turbulent region diminishes as thedistance downstream of the obstruction increases, the most criticalsituation is when the obstruction is relatively close to the NWS 10 msensor. Furthermore, if the wind approaching the obstruction is at somepositive angle relative to the ground surface, the height of theturbulent region may be significantly greater than the obstructionitself. Suppose the height of an obstruction close to the NWS 10 msensor is 25 ft (7.62 m). Conceivably, the height of the turbulentregion might be 10 ft (3.1 m) greater at 35 ft (10.67 m), such that asensor at the 33 ft level would be immersed in the turbulence. Thespecification that the sensor be “at least 15 ft (4.57 m) above theheight of any obstruction” would, in this case, avoid that possibility.Similar reasoning could be made for the case of an obstruction outsidethe 500 ft radius but inside the 1000 ft radius. It appears that theabove guidelines were developed so as to insure that the NWS 10 m sensorwould be exposed to free-stream conditions at all times even forrelatively larger obstructions.

The ASOS/AWOS specifications on the NWS 10 m sensor guarantee that thefree-stream velocity will be measured at the sensor location for allvelocities providing the wind/surface contact distance (l) is 1000 ft orless. But this wind/surface contact distance may be much greater thanthe 1000 ft in cases where the sensor is located alongside an airport'srunway. An example is the 6,500 ft (1,981.22 m) runway at the ItascaCounty Airport in Grand Rapids, Minn. where the sensor is located at adistance of 35 ft (10.67 m) from the side of the runway and about 5,000ft (1,524 m) from one end of the runway. The runway would constitute asmooth surface such that Eq. (34) would correctly apply. If a 20 mphwind in the direction of the runway approaches the sensor from the 5,000ft end of the runway, the boundary layer thickness at the sensorlocation is δ=29.5 ft (8.99 m). When V=70 mph, δ=22.9 ft (6.98 m). Thismeans that over an expected 20 mph to 70 mph velocity range, the sensorwould always be exposed to free-stream conditions. However, this wouldno longer be true if V≦11.3 mph (18.19 Km/hr) because when V=11.3 mph, δis exactly equal to 33 ft. For example, when V=8 mph (12.87 Km/hr),δ=35.4 ft (10.79 m) and the sensor would lie 2.4 ft (0.73 m) below theboundary layer edge. From Eq. (35), the velocity (v) being recorded bythis submerged sensor when V=8 mph, y=33 ft, δ=35.4 ft, and assumingn=9, would be 7.94 mph (12.78 Km/hr). This is within 1% of the actualfree-stream value and would not result in any error of consequence inwind chill predictions. Because this particular sensor is probablyrepresentative of all sensors nationwide, the conclusion is made thatall sensors, at all times, record a velocity that is very close to orexactly the free-stream velocity.

The discussion so far has dealt with wind reduction in instances wherethe head is immersed in a wind-generated turbulent boundary layer. Thesecases represent situations that are relatively simple to analyze. In theother cases, where the individual is within the turbulent regiondownstream of an obstruction, the WRF is more difficult to determine.Only when the individual is very close to the obstruction and totallywithin the flow separation region (x) as shown in FIG. 30 can it besafely said that he is completely shielded from the wind, in which caseWRF=1. This is the only value of WRF that is clearly defined whenconsidering obstructions. As the individual in the separation regionmoves away from the obstruction towards the wind reattachment point (x),the WRF will decrease becoming WRF=0 before reaching x. Because the WRFvaries from 0 to 1.0 over the separation region, there is likely to beat least one location within this region where WRF=0.33. This is theonly other instance where the current use of WRF=0.33 would correctlyapply.

Individuals in an urban area may be subjected to a combined effect of aboundary layer and a separated region. An individual located at somedistance downstream from an obstruction on a clear street with buildingson either side may experience a boundary layer. If the wind/surfacedistance (l) along the street is 1,500 ft (457.21 m) or less, which iscomparable to about two standard city blocks, then from FIG. 33,WRF<0.15. As stated previously, in situations like this, it would makesense to assume WRF=0, and in so doing provide a measure of safety inthe wind chill prediction. Now if vehicle signs, lamp posts, and otherobstructions exist along the street, the individual may not experiencean uninterrupted boundary layer, but rather an exposure to multipleturbulent regions. Determining a WRF here would be nearly impossible.Unless the individual manages to become completely sheltered, in whichcase the WRF=1, the actual wind reduction would be quite small if theseturbulent regions are, relatively speaking, far from the individual.Again, assuming WRF=0 would be the preferred choice.

There are two situations where a wind reduction at head level is theresult of a modification of the NWS 10 m velocity value and not theresult of the individual being exposed to a wind-generated boundarylayer. The first is an increase in the NWS 10 m free-stream value thatcould occur in a large urban area as a result of what Schwerdt referredto as “air funneling around tall buildings.” This increase could becomputed knowing the size, number, and layout of the buildings. Thesecond refers to a case where the NWS 10 m value of the velocity wouldbe decreased. Picture the previous illustration of the wind blowing downthe 1,500 ft length of street and then being deflected 900 around abuilding and continuing to flow down a cross street. The deflected wind,including its boundary layer, would generate a turbulent region on thecross street side of the building, with a subsequent reattachment to thesurface at a distance (x) downstream of the turning point. Energy lossesincurred by the flow as a result of this turning would be reflected as areduced value in the free-stream velocity after the turn, as comparedwith its NWS 10 m value. As before, at distances of l<l_(max) downstreamof the reattachment point (x), WRF=0 from FIG. 32. Therefore, in thesecases there would be no wind reduction due to boundary layer immersion.However, there would be a reduction due to a decrease in the velocity ofthe reattached flow. This decrease depends upon the magnitude of thevelocity (V) before the turn and the radius of the turning flow aroundthe obstruction. In this instance, the decrease in velocity due toturning should be accounted for, but its actual determination might bedifficult. Couple this situation with the possibility of the turned flowbeing accelerated by the above-mentioned funneling, and there is apossibility that the velocity of the turned flow may come back to itsoriginal NWS 10 m value, or even exceed it. In that case, the actualvalue of the velocity at head level may not be too different from theNWS 10 m value. In this situation, where the conventional WRF=0 as faras the wind/surface boundary layer is concerned, there might be a windreduction or even a wind increase at head level due to changes in themagnitude of the NWS 10 m value. Unless absolutely necessary in aspecific evaluation of wind chill within the confines of a city, it issuggested that the actual value of the NWS 10 m value be used at alltimes.

Finally, there are cases where an individual is exposed to an equivalentwind speed due to his motion even when calm wind (V=0 mph) conditionsprevail. These cases include motorcycle riders, snowmobile riders, anddownhill skiers. Facial freezing could occur at low ambient temperaturesand high speeds of motion in the absence of facial protection such as ahelmet. In each case, there is no boundary layer and consequently nowind reduction, therefore, WRF=0 at all times.

Based on the above discussions, the following statements and commentscan be made concerning the concept of wind reduction: The WRF=0.33corresponding to the currently assumed 50% reduction in the NWS 10 mwind speed value is correct only if (a) the wind-generated boundarylayer thickness during the Saskatoon velocity profile measurements wasδ=33 ft (10.1 m) and if (b) the same boundary layer thickness exists atthe location of the individual being considered. This is a singular andhighly unlikely situation. It emphasizes the fact that the usage ofWRF=0.33 for all calculations of the wind chill temperature, withoutregard for the individual's position relative to the wind, isinaccurate. A WRF=0.33 will correctly and uniquely apply to anindividual at some point within the separation region on the leewardside of an obstruction. This is the only instance, in realisticsituations, in which the current wind chill model would come close topredicting the correct wind chill temperature (T_(wc)). A WRF=1.0represents the special case where the individual in the above separationregion is adjacent to the obstruction such that the latter completelyblocks him from the oncoming flow.

Calculations of boundary layer values of the WRF as a function of thevelocity (V) and an individual's distance (D) from an obstructionassuming Steadman's value of the velocity profile exponent (1/4.76),shows that for: (a) V=20 mph: (1) WRF=0, for D<600 ft and (2) WRF−0.055(average) for 600 ft<D<1,500 ft. For (b) V=70 mph: (1) WRF=0, for D<850ft and (2) WRF=0.072 (average) for 850 ft<D<1,500 ft.

The distance range, 600 ft<D<850 ft, may reflect situations in an urbanarea, while the range 600 ft<D<1,500 ft might be considered as those ina more rural area. Since it is anticipated that these two areasencompass more than 95% of the populace, and since the WRF values arevery small or non-existent, it emphasizes the importance and need toremove the equivalent correction (WRF=0.33) used in the current windchill model.

The above calculations of WRF were based on Steadman's velocity profileexponent (1/4.76). If the usage of this exponent is found to beincorrect, then a more appropriate one of lower value (⅛, 1/9, 1/10,etc.), which is consistent with higher velocities and Reynolds numbers,should be used. Decreasing the value of this exponent decreases thevalues of WRF even further; this emphasizes the advisability of assumingWRF=0 when catering to the public at large.

It is suggested that the actual NWS 10 m velocity value be used at alltimes, although the local value of the free-stream velocity in an urbanenvironment may be somewhat larger or smaller. And finally, the WRF=0for all motion-driven sport activists such as cyclists and skiers undercalm conditions (V=0 mph), because of the absence of a wind-generatedboundary layer.

The above results show that, except for a few special cases, the WRF=0or is only slightly greater than zero for almost everybody in thelistening area. Where WRF takes on small positive values of D between600 ft and 1,500 ft from an obstruction, the recommendation is made thatWRF=0 in this region also. Doing so could, in cases where facialfreezing is a possibility, provide the listener with a degree of safetyby informing him of a time to freeze before it actually occurs. Forvalues of D greater than 1,500 ft, WRF may become quite large as shownin FIG. 33. In these cases, WRF should be considered in the calculationof the wind chill temperature (T_(wc)).

The inventors suggest the following recommendations be considered forimplementation of the wind chill model of the present invention toprovide the listening public with a more accurate wind chill temperature(T_(wc)) at the moment of exposure, facial temperature (T_(f)) at anytime during the exposure and the time to freeze (t_(f)): (1) Determinethe latitude (LAT) for the entire listening area from Table 2; (2)Determine the average altitude (H) of the listening area; (3) Determinefrom Table 1 the most likely physical activity (M_(act)) of theinhabitants within the listening area. For large urban or rural areas,the most likely activity would probably be “walking about” so thatM_(act)=31 Btu/hr-ft² (97.79 W/m²); (4) With the likelihood that 95% ofthe people in the listening area will experience a wind reduction factor(WRF) equal to or less than 0.072, assume WRF=0. This is not onlyconvenient, but also beneficial to the listeners, since it affords themadded safety by informing them of a time to freeze (t_(f)) before itactually occurs. For other situations where individuals are at a knowndistance from an obstruction, the WRF could be determined and used inthe calculations; (5) Unless sunshine is a certainty over the entirelistening area and for the entire prediction period, assume G=0Btu/hr-ft² (0 W/m²). This too provides the listener with a degree ofsafety by again informing him of a time to freeze (t_(f)) before itactually occurs. If sunshine is expected to be present over the entireprediction period, G is determined from Table 2; and (6) Themeteorologist needs to clarify to the public the real meaning of windchill. Years of mystifying statements on the subject, such as theerroneous statement that the reason we experience wind chill is duesolely to evaporation from the skin surface, has left the publicconfused. This confusion still reigns as evidenced by a recent (2005)statement by a well-known network TV news commentator that wind chill isa “phony number”, Andy Rooney of CBS 60 min., January 2005. The publicneeds to be informed that wind chill is not just seasonal, but rather ayear-round phenomenon.

The methods and procedures for calculating wind chill temperature,T_(wc), equivalent temperature, T_(eq), time to freeze, t_(f), facialtemperature, T_(f) ^(m+Δt), as a function of time, altitude correctionfactor, Δt_(f)/1000, and any other related calculations as disclosedherein, may be practiced as a computer program, i.e.,computer-executable instructions, suitable for processing by a processorbased on selected input variables. Such computer-executable instructionsmay, of course be stored on any suitable computer-readable medium. FIG.35 is a block diagram of an embodiment of a computer readable medium 100suitable for storing computer-executable instructions 102 forcalculating wind chill temperature, T_(wc), equivalent temperature,T_(eq), and time to freeze, t_(f), and any other related calculationsaccording to the present invention.

FIG. 36 is a system 200 for determining wind chill temperature, T_(wc),equivalent temperature, T_(eq), time to freeze, t_(f), facialtemperature, T_(f) ^(m+Δt), as a function of time, altitude correctionfactor, Δt_(f)/1000, and any other related calculations disclosed hereinaccording to the present invention. System 200 may include an inputdevice 202, an output device 204, a memory device 206, and a processor208 in communication with the input device 202, the output device 204,and the memory device 206. The processor 208 may be configured toexecute computer-readable instructions 210 stored on the memory device206. The memory device 206 may further include computer-readableinstructions 210 for implementing any of the methods disclosed herein.According to one embodiment of system 200, the memory device 206 mayinclude computer-readable instructions 210 for implementing a method fordetermining the wind chill temperature, T_(wc), according to the methodsof the present invention. According to another embodiment of system 200,the memory device 206 may include computer-readable instructions 210 forimplementing a method for determining an equivalent temperature, T_(eq),of a two-dimensional object, according to the methods of the presentinvention. According to still another embodiment of system 200, thememory device 206 may include computer-readable instructions 210 forimplementing a method for determining time to freeze, t_(f), accordingto the methods of the present invention. Of course, one of skill in theart will recognize that system 200 may be used to calculate selected orall of the wind chill related parameters disclosed herein.

While the foregoing advantages of the present invention are manifestedin the detailed description and illustrated embodiments of theinvention, a variety of changes can be made to the configuration,design, and construction of the invention to achieve those advantages.Hence, reference herein to specific details of the structure andfunction of the present invention is by way of example only and not byway of limitation.

1. A method for determining an equivalent temperature, T_(eq), of atwo-dimensional object, comprising calculating,$T_{eq} = {T_{s} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{{\frac{0.00963\left( {P\; V} \right)^{0.5}}{\left\lbrack {0.5\left( {T_{s} + T_{a}} \right)} \right\rbrack^{0.04}L^{0.5}}\left( {T_{s} - T_{a}} \right)} +} \\{{\sigma\; ɛ\left( {T_{s}^{4} - T_{a}^{4}} \right)} + {\overset{.}{w}\; l_{e}} - {\alpha\; G} - {\frac{k}{s}\left( {T_{b} - T_{s}} \right)}}\end{bmatrix}} \right\}^{1/{({1 + \phi})}}}$ wherein T_(s) comprisessurface temperature, L comprises length of the object, φ comprises afunction of heat source, C₁ comprises a function of the surfacetemperature and the ambient temperature, P comprises ambient pressure, Vcomprises free-stream wind velocity, T_(a) comprises ambienttemperature, σ comprises Stefan-Boltzmann constant, ε comprisesemissivity of the object, {dot over (w)} comprises water evaporationflux rate, l_(e) comprises latent heat of evaporation, α comprisesthermal absorptivity, G comprises insolation value, k comprises thermalconductivity, s comprises thickness of the object and T_(b) comprisesobject internal temperature.
 2. The method according to claim 1, whereinthe equivalent temperature, T_(eq), further comprises:${T_{eq}\left( {V = 0} \right)} = {T_{s} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{{ɛ\;{\sigma\left( {T_{s}^{4} - T_{a}^{4}} \right)}} +} \\{{\frac{k}{s}\left( {T_{b} - T_{s}} \right)} + {\alpha\; G} - {\overset{.}{w}\; l_{e}}}\end{bmatrix}} \right\}^{{1/}{({1 + \phi})}}}$ when the wind velocity,V, is approximately zero.
 3. A computer-readable medium storingcomputer-executable instructions for performing the method according toclaim
 1. 4. A method for determining facial temperature, T_(f) ^(m+Δt),as a function of time, comprising calculating,$T_{f}^{m + {\Delta\; t}} = {T_{f}^{m} + {\left( \frac{2}{\rho\; C_{p}s} \right){\left( {\Delta\; t} \right)\begin{bmatrix}{{K\left( {T_{CR} - T_{f}^{m}} \right)} + {\alpha\; G} -} \\{\frac{(1.8062)\left\{ {\left( {1 - {WRF}} \right){V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}} \right\}^{0.5}}{{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.04}\; D^{0.5}} -}\left( {T_{f} - T_{a}} \right)} \\{{\sigma\;{ɛ\left( {T_{f} + T_{a}} \right)}\left( {T_{f}^{2} + T_{a}^{2}} \right)\left( {T_{f} - T_{a}} \right)} - {\overset{.}{w}\; l_{e}}}\end{bmatrix}}}}$ wherein T_(f) ^(m) comprises initial facialtemperature, ρ comprises skin density, C_(p) comprises skin's specificheat at constant pressure, s comprises total skin thickness, Δtcomprises time increment, K comprises thermal conductance, T_(CR)comprises core temperature, α comprises thermal absorptivity, Gcomprises insolation value, WRF comprises wind reduction factor, Vcomprises wind velocity at NWS 10 m level, H comprises locationaltitude, T_(f) comprises facial temperature, T_(a) comprises ambienttemperature, D comprises head diameter, σ comprises Stefan-Boltzmannconstant, ε comprises skin's emissivity, {dot over (w)} comprises waterevaporation flux rate and l_(e) comprises latent heat of evaporation. 5.A computer-readable medium storing computer-executable instructions forperforming the method according to claim
 4. 6. A method for determiningtime to freeze, t_(f), comprising calculating$t_{f} = \frac{\left\{ {a + {b\mspace{11mu}{\ln(V)}} + {c\left\lbrack {\ln(V)} \right\rbrack}^{2} + {d\left\lbrack {\ln(V)} \right\rbrack}^{3} + {e\; T_{a}}} \right\}}{\left\{ {1 + {f\mspace{11mu}{\ln(V)}} + {g\left\lbrack {\ln(V)} \right\rbrack}^{2} + {h\; T_{a}} + {i\left( T_{a} \right)}^{2}} \right\}}$wherein V comprises wind velocity at NWS 10 m level, T_(a) comprisesambient temperature, α comprises approximately 12.2, b comprisesapproximately −4.73, c comprises approximately 0.714, d comprisesapproximately −0.0404, e comprises approximately 0.00203, f comprisesapproximately −0.0428, g comprises approximately 0.00480, h comprisesapproximately −0.0162 and i comprises approximately −2.37×10⁻⁵.
 7. Themethod for determining time to freeze, t_(f), according to claim 6,further comprising calculating an altitude correction factor,Δt_(f)/1000, $\begin{matrix}{\frac{\Delta\; t_{f}}{1000} = {a + {b\mspace{11mu}{\ln(V)}} + {c\; T_{a}} + {d\left\lbrack {\ln(V)} \right\rbrack}^{2} + {e\; T_{a}^{2}} + {f\; T_{a}{\ln(V)}} + {g\left\lbrack {\ln(V)} \right\rbrack}^{3} + {h\; T_{a}^{3}} + {i\; T_{a}^{2}{\ln(V)}} + {j\;{T_{a}\left\lbrack {\ln(V)} \right\rbrack}^{2}}}} & \;\end{matrix}$ wherein α comprises approximately 0.170, b comprisesapproximately −0.0377, c comprises approximately 0.00118, d comprisesapproximately 0.00225, e comprises approximately 6.16×10⁻⁶, f comprisesapproximately −9.66×10⁻⁵, g comprises approximately 3.34×10⁻⁶, hcomprises approximately 1.79×10⁻⁸, i comprises approximately 1.21×10⁻⁹,j comprises approximately −1.41×10⁻⁷; and adding Δt_(f)/1000 to t_(f)for each 1000 feet above sea level.
 8. A computer-readable mediumstoring computer-executable instructions for performing the methodaccording to claim
 7. 9. A computer-readable medium storingcomputer-executable instructions for performing a method for determiningwind chill temperature, T_(wc), the method comprising: calculating$T_{wc} = {T_{f} - \left\{ {\frac{L^{\phi}}{C_{1}}\begin{bmatrix}{\frac{(1.8062)\left\{ {\left( {1 - {WRF}} \right){V\left\lbrack {1 - {\left( {6.92 \times 10^{- 6}} \right)H}} \right\rbrack}^{5.21}} \right\}^{0.5}}{{\left\lbrack {0.5\left( {T_{f} + T_{a}} \right)} \right\rbrack^{0.04}\; D^{0.5}} +}\left( {T_{f} - T_{a}} \right)} \\{{\sigma\; ɛ\left( {T_{f} + T_{a}} \right)\left( {T_{f}^{2} + T_{a}^{2}} \right)\left( {T_{f} - T_{a}} \right)} +} \\{{\overset{.}{w}\; l_{e}} - {\alpha\; G} - {K\left( {T_{CR} - T_{f}} \right)}}\end{bmatrix}} \right\}^{1/{({1 + \phi})}}}$ wherein T_(f) comprisesfacial temperature, L comprises human face length, φ comprises afunction of heat source, C₁ comprises a function of the facialtemperature and the ambient temperature, WRF comprises wind reductionfactor, V comprises wind velocity at NWS 10 m level, H compriseslocation altitude, T_(a) comprises ambient temperature, D comprises headdiameter, σ comprises Stefan-Boltzmann constant, ε comprises skinemissivity, {dot over (w)} comprises water evaporation flux rate, l_(e)comprises latent heat of evaporation, α comprises thermal absorptivity,G comprises insolation value, K comprises thermal conductance and T_(CR)comprises core temperature.